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Khan Academy - Linear Algebra 1-48

Distance between planes | Vectors and spaces | Linear Algebra | Khan Academy
Distance between planes | Vectors and spaces | Linear Algebra | Khan Academy2010 IIT JEE Paper 1 Problem 51 Distance Between PlanesWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/matrices_elimination/v/…
- 28 Dec 10, 11:29pm -
Point distance to plane | Vectors and spaces | Linear Algebra | Khan Academy
Point distance to plane | Vectors and spaces | Linear Algebra | Khan AcademyDistance between a point and a plane in three dimensionsWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/dot_cross_products/v/d…
- 28 Dec 10, 11:07pm -
Normal vector from plane equation | Vectors and spaces | Linear Algebra | Khan Academy
Normal vector from plane equation | Vectors and spaces | Linear Algebra | Khan AcademyFiguring out a normal vector to a plane from its equationWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/vectors_and_spaces/dot_cross_p…
- 28 Dec 10, 10:30pm -
Vector triple product expansion (very optional) | Vectors and spaces | Linear Algebra | Khan Academy
Vector triple product expansion (very optional) | Vectors and spaces | Linear Algebra | Khan AcademyA shortcut for having to evaluate the cross product of three vectorsWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/vecto…
- 28 Dec 10, 5:26pm -
Showing that an eigenbasis makes for good coordinate systems | Linear Algebra | Khan Academy
Showing that an eigenbasis makes for good coordinate systems | Linear Algebra | Khan AcademyShowing that an eigenbasis makes for good coordinate systemsThis is the last video in our linear algebra subject! Now move on to our first video in recr…
- 15 Nov 09, 6:42pm -
Eigenvectors and eigenspaces for a 3x3 matrix | Linear Algebra | Khan Academy
Eigenvectors and eigenspaces for a 3x3 matrix | Linear Algebra | Khan AcademyEigenvectors and eigenspaces for a 3x3 matrixWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/eigen_everything/v/linear-algebra-s…
- 14 Nov 09, 5:25pm -
Eigenvalues of a 3x3 matrix | Alternate coordinate systems (bases) | Linear Algebra | Khan Academy
Eigenvalues of a 3x3 matrix | Alternate coordinate systems (bases) | Linear Algebra | Khan AcademyDetermining the eigenvalues of a 3x3 matrixWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/eigen_everything…
- 14 Nov 09, 4:53pm -
Finding eigenvectors and eigenspaces example | Linear Algebra | Khan Academy
Finding eigenvectors and eigenspaces example | Linear Algebra | Khan AcademyFinding the eigenvectors and eigenspaces of a 2x2 matrixWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/eigen_everything/v/linear…
- 14 Nov 09, 2:44am -
Example solving for the eigenvalues of a 2x2 matrix | Linear Algebra | Khan Academy
Example solving for the eigenvalues of a 2x2 matrix | Linear Algebra | Khan AcademyExample solving for the eigenvalues of a 2x2 matrixWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/eigen_everything/v/line…
- 14 Nov 09, 12:20am -
Proof of formula for determining eigenvalues | Linear Algebra | Khan Academy
Proof of formula for determining eigenvalues | Linear Algebra | Khan AcademyProof of formula for determining EigenvaluesWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/eigen_everything/v/linear-algebra-exa…
- 14 Nov 09, 12:11am -
Introduction to eigenvalues and eigenvectors | Linear Algebra | Khan Academy
Introduction to eigenvalues and eigenvectors | Linear Algebra | Khan AcademyWhat eigenvectors and eigenvalues are and why they are interestingWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/eigen_everythin…
- 13 Nov 09, 11:34pm -
Gram-Schmidt example with 3 basis vectors | Linear Algebra | Khan Academy
Gram-Schmidt example with 3 basis vectors | Linear Algebra | Khan AcademyGram-Schmidt example with 3 basis vectorsWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/eigen_everything/v/linear-algebra-introduct…
- 13 Nov 09, 8:06pm -
Gram-Schmidt process example | Alternate coordinate systems (bases) | Linear Algebra | Khan Academy
Gram-Schmidt process example | Alternate coordinate systems (bases) | Linear Algebra | Khan AcademyUsing Gram-Schmidt to find an orthonormal basis for a plane in R3Watch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate…
- 13 Nov 09, 7:40pm -
The Gram-Schmidt process | Alternate coordinate systems (bases) | Linear Algebra | Khan Academy
The Gram-Schmidt process | Alternate coordinate systems (bases) | Linear Algebra | Khan AcademyFinding an orthonormal basis for a subspace using the Gram-Schmidt ProcessWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alte…
- 13 Nov 09, 7:13pm -
Orthogonal matrices preserve angles and lengths | Linear Algebra | Khan Academy
Orthogonal matrices preserve angles and lengths | Linear Algebra | Khan AcademyShowing that orthogonal matrices preserve angles and lengthsWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/orthonormal_basis/…
- 13 Nov 09, 6:30pm -
Example using orthogonal change-of-basis matrix to find transformation matrix | Khan Academy
Example using orthogonal change-of-basis matrix to find transformation matrix | Khan AcademyExample using orthogonal change-of-basis matrix to find transformation matrixWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alt…
- 13 Nov 09, 4:18pm -
Finding projection onto subspace with orthonormal basis example | Linear Algebra | Khan Academy
Finding projection onto subspace with orthonormal basis example | Linear Algebra | Khan AcademyExample of finding the transformation matrix for the projection onto a subspace with an orthonormal basisWatch the next lesson: https://www.khanacade…
- 12 Nov 09, 11:44pm -
Projections onto subspaces with orthonormal bases | Linear Algebra | Khan Academy
Projections onto subspaces with orthonormal bases | Linear Algebra | Khan AcademyProjections onto subspaces with orthonormal basesWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/orthonormal_basis/v/lin-alg…
- 12 Nov 09, 11:44pm -
Coordinates with respect to orthonormal bases | Linear Algebra | Khan Academy
Coordinates with respect to orthonormal bases | Linear Algebra | Khan AcademySeeing that orthonormal bases make for good coordinate systemsWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/orthonormal_basis/…
- 12 Nov 09, 10:50pm -
Introduction to orthonormal bases | Linear Algebra | Khan Academy
Introduction to orthonormal bases | Linear Algebra | Khan AcademyLooking at sets and bases that are orthonormal -- or where all the vectors have length 1 and are orthogonal to each other.Watch the next lesson: https://www.khanacademy.org/math/l…
- 12 Nov 09, 9:43pm -
Changing coordinate systems to help find a transformation matrix | Linear Algebra | Khan Academy
Changing coordinate systems to help find a transformation matrix | Linear Algebra | Khan AcademyChanging our coordinate system to find the transformation matrix with respect to standard coordinatesWatch the next lesson: https://www.khanacademy.…
- 12 Nov 09, 7:03pm -
Alternate basis transformation matrix example part 2 | Linear Algebra | Khan Academy
Alternate basis transformation matrix example part 2 | Linear Algebra | Khan AcademyShowing that the transformation matrix with respect to basis B actually works. Brief point on why someone would want to operate in a different basis to begin with…
- 12 Nov 09, 4:44pm -
Alternate basis transformation matrix example | Linear Algebra | Khan Academy
Alternate basis transformation matrix example | Linear Algebra | Khan AcademyExample of finding the transformation matrix for an alternate basisWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/change_of_ba…
- 12 Nov 09, 12:21am -
Transformation matrix with respect to a basis | Linear Algebra | Khan Academy
Transformation matrix with respect to a basis | Linear Algebra | Khan AcademyFinding the transformation matrix with respect to a non-standard basisWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/change_of_…
- 11 Nov 09, 8:57pm -
Invertible change of basis matrix | Linear Algebra | Khan Academy
Invertible change of basis matrix | Linear Algebra | Khan AcademyUsing an invertible change of basis matrix to go between different coordinate systemsWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/change_…
- 11 Nov 09, 7:06pm -
Change of basis matrix | Alternate coordinate systems (bases) | Linear Algebra | Khan Academy
Change of basis matrix | Alternate coordinate systems (bases) | Linear Algebra | Khan AcademyUsing a change of basis matrix to get us from one coordinate system to another.Watch the next lesson: https://www.khanacademy.org/math/linear-algebra/a…
- 11 Nov 09, 6:33pm -
Coordinates with respect to a basis | Linear Algebra | Khan Academy
Coordinates with respect to a basis | Linear Algebra | Khan AcademyUnderstanding alternate coordinate systemsWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/change_of_basis/v/linear-algebra-change-of-basis…
- 11 Nov 09, 5:54pm -
Another least squares example | Alternate coordinate systems (bases) | Linear Algebra | Khan Academy
Another least squares example | Alternate coordinate systems (bases) | Linear Algebra | Khan AcademyUsing least squares approximation to fit a line to pointsWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/…
- 10 Nov 09, 11:27pm -
Least squares examples | Alternate coordinate systems (bases) | Linear Algebra | Khan Academy
Least squares examples | Alternate coordinate systems (bases) | Linear Algebra | Khan AcademyAn example using the least squares solution to an unsolvable systemWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bas…
- 10 Nov 09, 11:27pm -
Least squares approximation | Linear Algebra | Khan Academy
Least squares approximation | Linear Algebra | Khan AcademyThe least squares approximation for otherwise unsolvable equationsWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/orthogonal_projections/v/linear-…
- 10 Nov 09, 10:06pm -
Projection is closest vector in subspace | Linear Algebra | Khan Academy
Projection is closest vector in subspace | Linear Algebra | Khan AcademyShowing that the projection of x onto a subspace is the closest vector in the subspace to xWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_…
- 10 Nov 09, 9:24pm -
Another example of a projection matrix | Linear Algebra | Khan Academy
Another example of a projection matrix | Linear Algebra | Khan AcademyFiguring out the transformation matrix for a projection onto a subspace by figuring out the matrix for the projection onto the subspace's orthogonal complement firstWatch the…
- 10 Nov 09, 7:42pm -
Subspace projection matrix example | Linear Algebra | Khan Academy
Subspace projection matrix example | Linear Algebra | Khan AcademyExample of a transformation matrix for a projection onto a subspaceWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/orthogonal_projections/v…
- 10 Nov 09, 6:28pm -
A projection onto a subspace is a linear transformation | Linear Algebra | Khan Academy
A projection onto a subspace is a linear transformation | Linear Algebra | Khan AcademyShowing that a projection onto a subspace is a linear transformationWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/o…
- 10 Nov 09, 5:58pm -
Visualizing a projection onto a plane | Linear Algebra | Khan Academy
Visualizing a projection onto a plane | Linear Algebra | Khan AcademyVisualizing a projection onto a plane. Showing that the old and new definitions of projections aren't that different.Watch the next lesson: https://www.khanacademy.org/math/li…
- 10 Nov 09, 5:24pm -
Projections onto subspaces | Linear Algebra | Khan Academy
Projections onto subspaces | Linear Algebra | Khan AcademyProjections onto subspacesWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/orthogonal_projections/v/linear-alg-visualizing-a-projection-onto-a-plane…
- 10 Nov 09, 4:53pm -
Showing that A-transpose x A is invertible | Matrix transformations | Linear Algebra | Khan Academy
Showing that A-transpose x A is invertible | Matrix transformations | Linear Algebra | Khan AcademyShowing that (transpose of A)(A) is invertible if A has linearly independent columnsWatch the next lesson: https://www.khanacademy.org/math/linea…
- 9 Nov 09, 11:34pm -
Rowspace solution to Ax = b example | Linear Algebra | Khan Academy
Rowspace solution to Ax = b example | Linear Algebra | Khan AcademyVisualizing the rowspace solution to Ax=bWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/orthogonal_projections/v/linear-algebra-projectio…
- 9 Nov 09, 10:48pm -
Rowspace solution to Ax = b example | Linear Algebra | Khan Academy
Rowspace solution to Ax = b example | Linear Algebra | Khan AcademyVisualizing the rowspace solution to Ax=bWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/orthogonal_projections/v/linear-algebra-projectio…
- 9 Nov 09, 8:31pm -
Unique rowspace solution to Ax = b | Linear Algebra | Khan Academy
Unique rowspace solution to Ax = b | Linear Algebra | Khan AcademyShowing that, for any b that is in the column space of A, there is a unique member of the row space that is the "smallest" solution to Ax=bWatch the next lesson: https://www.khan…
- 9 Nov 09, 5:54pm -
Orthogonal complement of the nullspace | Linear Algebra | Khan Academy
Orthogonal complement of the nullspace | Linear Algebra | Khan AcademyThe orthogonal complement of the nullspace and left nullspace.Watch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/othogonal_complements/v/l…
- 9 Nov 09, 4:29pm -
Orthogonal complement of the orthogonal complement | Linear Algebra | Khan Academy
Orthogonal complement of the orthogonal complement | Linear Algebra | Khan AcademyFinding that the orthogonal complement of the orthogonal complement of V is VWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_base…
- 9 Nov 09, 4:29pm -
Representing vectors in rn using subspace members | Linear Algebra | Khan Academy
Representing vectors in rn using subspace members | Linear Algebra | Khan AcademyShowing that any member of Rn can be represented as a unique sum of a vector in subspace V and a vector in the orthogonal complement of V.Watch the next lesson: ht…
- 8 Nov 09, 10:36pm -
dim(v) + dim(orthogonal complement of v) = n | Linear Algebra | Khan Academy
dim(v) + dim(orthogonal complement of v) = n | Linear Algebra | Khan AcademyShowing that if V is a subspace of Rn, then dim(V) + dim(V's orthogonal complement) = nWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_…
- 7 Nov 09, 12:30am -
rank(a) = rank(transpose of a) | Matrix transformations | Linear Algebra | Khan Academy
rank(a) = rank(transpose of a) | Matrix transformations | Linear Algebra | Khan AcademyRank(A) = Rank(transpose of A)Watch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/matrix_transpose/v/lin-alg-showin…
- 7 Nov 09, 12:30am -
Orthogonal complements | Alternate coordinate systems (bases) | Linear Algebra | Khan Academy
Orthogonal complements | Alternate coordinate systems (bases) | Linear Algebra | Khan AcademyOrthogonal Complements as subspaces.Watch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/othogonal_complements/v/line…
- 6 Nov 09, 5:54pm -
Visualizations of left nullspace and rowspace | Linear Algebra | Khan Academy
Visualizations of left nullspace and rowspace | Linear Algebra | Khan AcademyRelationship between left nullspace, rowspace, column space and nullspace.Watch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations…
- 6 Nov 09, 5:18pm -
Rowspace and left nullspace | Matrix transformations | Linear Algebra | Khan Academy
Rowspace and left nullspace | Matrix transformations | Linear Algebra | Khan AcademyRowspace and Left NullspaceWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/matrix_transpose/v/lin-alg-visualizatio…
- 6 Nov 09, 5:18pm -

Khan Academy - Linear Algebra 49-97

Transpose of a matrix product | Matrix transformations | Linear Algebra | Khan Academy
Transpose of a matrix product | Matrix transformations | Linear Algebra | Khan AcademyTaking the transpose of the product of two matricesWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/matrix_transp…
- 28 Jul 11, 6:40pm -
Transpose of a vector | Matrix transformations | Linear Algebra | Khan Academy
Transpose of a vector | Matrix transformations | Linear Algebra | Khan AcademyTranspose of a column vector. Matrix-matrix products using vectorsWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/matrix…
- 5 Nov 09, 5:26pm -
Transposes of sums and inverses | Matrix transformations | Linear Algebra | Khan Academy
Transposes of sums and inverses | Matrix transformations | Linear Algebra | Khan AcademyTransposes of sums and inversesWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/matrix_transpose/v/linear-algeb…
- 5 Nov 09, 2:47pm -
Determinant of transpose | Matrix transformations | Linear Algebra | Khan Academy
Determinant of transpose | Matrix transformations | Linear Algebra | Khan AcademyProof by induction that transposing a matrix does not change its determinantWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transform…
- 5 Nov 09, 2:47pm -
Transpose of a matrix | Matrix transformations | Linear Algebra | Khan Academy
Transpose of a matrix | Matrix transformations | Linear Algebra | Khan AcademyTranspose of a matrixWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/matrix_transpose/v/linear-algebra-determinant-of-tr…
- 4 Nov 09, 9:00pm -
Determinant as scaling factor | Matrix transformations | Linear Algebra | Khan Academy
Determinant as scaling factor | Matrix transformations | Linear Algebra | Khan AcademyViewing the determinant of the transformation matrix as a scaling factor of regionsWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matr…
- 4 Nov 09, 7:16pm -
Determinant and area of a parallelogram | Matrix transformations | Linear Algebra | Khan Academy
Determinant and area of a parallelogram | Matrix transformations | Linear Algebra | Khan AcademyRealizing that the determinant of a 2x2 matrix is equal to the area of the parallelogram defined by the column vectors of the matrixWatch the next l…
- 4 Nov 09, 5:38pm -
Simpler 4x4 determinant | Matrix transformations | Linear Algebra | Khan Academy
Simpler 4x4 determinant | Matrix transformations | Linear Algebra | Khan AcademyCalculating a 4x4 determinant by putting in in upper triangular form first.Watch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformat…
- 4 Nov 09, 1:36am -
Upper triangular determinant | Matrix transformations | Linear Algebra | Khan Academy
Upper triangular determinant | Matrix transformations | Linear Algebra | Khan AcademyThe determinant of an upper triangular matrixWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/determinant_depth/v/…
- 3 Nov 09, 11:27pm -
Determinant after row operations | Matrix transformations | Linear Algebra | Khan Academy
Determinant after row operations | Matrix transformations | Linear Algebra | Khan AcademyWhat happens to the determinant when we perform a row operationWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformation…
- 3 Nov 09, 10:52pm -
Duplicate row determinant | Matrix transformations | Linear Algebra | Khan Academy
Duplicate row determinant | Matrix transformations | Linear Algebra | Khan AcademyDeterminant of a matrix with duplicate rowsWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/determinant_depth/v/linea…
- 3 Nov 09, 10:52pm -
Determinant when row multiplied by scalar | Matrix transformations | Linear Algebra | Khan Academy
Determinant when row multiplied by scalar | Matrix transformations | Linear Algebra | Khan AcademyThe determinant when a row is multiplied by a scalarWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/…
- 3 Nov 09, 7:48pm -
Determinant when row is added | Matrix transformations | Linear Algebra | Khan Academy
Determinant when row is added | Matrix transformations | Linear Algebra | Khan AcademyThe determinant when one matrix has a row that is the sum of the rows of other matrices (and every other term is identical in the 3 matrices)Watch the next le…
- 3 Nov 09, 7:47pm -
(correction) scalar multiplication of row | Matrix transformations | Linear Algebra | Khan Academy
(correction) scalar multiplication of row | Matrix transformations | Linear Algebra | Khan AcademyCorrection of last video showing that the determinant when one row is multiplied by a scalar is equal to the scalar times the determinantWatch the…
- 3 Nov 09, 7:47pm -
Rule of Sarrus of determinants | Matrix transformations | Linear Algebra | Khan Academy
Rule of Sarrus of determinants | Matrix transformations | Linear Algebra | Khan AcademyA alternative "short cut" for calculating 3x3 determinants (Rule of Sarrus)Watch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_tran…
- 3 Nov 09, 12:35am -
Determinants along other rows/cols | Matrix transformations | Linear Algebra | Khan Academy
Determinants along other rows/cols | Matrix transformations | Linear Algebra | Khan AcademyFinding the determinant by going along other rows or columnsWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations…
- 3 Nov 09, 12:35am -
n x n determinant | Matrix transformations | Linear Algebra | Khan Academy
n x n determinant | Matrix transformations | Linear Algebra | Khan AcademyDefining the determinant for nxn matrices. An example of a 4x4 determinant.Watch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/i…
- 3 Nov 09, 12:34am -
3 x 3 determinant | Matrix transformations | Linear Algebra | Khan Academy
3 x 3 determinant | Matrix transformations | Linear Algebra | Khan AcademyDeterminants: Finding the determinant of a 3x3 matrixWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/inverse_of_matrices/v/l…
- 2 Nov 09, 10:10pm -
Formula for 2x2 inverse | Matrix transformations | Linear Algebra | Khan Academy
Formula for 2x2 inverse | Matrix transformations | Linear Algebra | Khan AcademyFiguring out the formula for a 2x2 matrix. Defining the determinant.Watch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/in…
- 2 Nov 09, 7:24pm -
Example of finding matrix inverse | Matrix transformations | Linear Algebra | Khan Academy
Example of finding matrix inverse | Matrix transformations | Linear Algebra | Khan AcademyExample of calculating the inverse of a matrixWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/inverse_of_mat…
- 31 Oct 09, 8:40pm -
Deriving a method for determining inverses | Matrix transformations | Linear Algebra | Khan Academy
Deriving a method for determining inverses | Matrix transformations | Linear Algebra | Khan AcademyDetermining a method for constructing inverse transformation matricesWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matri…
- 31 Oct 09, 8:40pm -
Showing that inverses are linear | Matrix transformations | Linear Algebra | Khan Academy
Showing that inverses are linear | Matrix transformations | Linear Algebra | Khan AcademyShowing that inverse transformations are also linearWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/inverse_o…
- 31 Oct 09, 8:40pm -
Simplifying conditions for invertibility | Matrix transformations | Linear Algebra | Khan Academy
Simplifying conditions for invertibility | Matrix transformations | Linear Algebra | Khan AcademyShowing that a transformation is invertible if and only if rref(A) is equal to the identity matrixWatch the next lesson: https://www.khanacademy.or…
- 31 Oct 09, 8:39pm -
Matrix condition for one-to-one trans | Matrix transformations | Linear Algebra | Khan Academy
Matrix condition for one-to-one trans | Matrix transformations | Linear Algebra | Khan AcademyShowing that the rank of the of an mxn transformation matrix has to be an for the transformation to be one-to-one (injective)Watch the next lesson: ht…
- 31 Oct 09, 8:39pm -
Exploring the solution set of Ax = b | Matrix transformations | Linear Algebra | Khan Academy
Exploring the solution set of Ax = b | Matrix transformations | Linear Algebra | Khan AcademyExploring the solution set of Ax=b (non homogeneous equations)Watch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformat…
- 31 Oct 09, 8:39pm -
Determining whether a transformation is onto | Linear Algebra | Khan Academy
Determining whether a transformation is onto | Linear Algebra | Khan AcademyDetermining whether a transformation is ontoWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/inverse_transformations/v/line…
- 31 Oct 09, 8:39pm -
Relating invertibility to being onto and one-to-one | Linear Algebra | Khan Academy
Relating invertibility to being onto and one-to-one | Linear Algebra | Khan AcademyRelating invertibility to being onto (surjective) and one-to-one (injective)Watch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transfo…
- 29 Oct 09, 8:26pm -
Surjective (onto) and injective (one-to-one) functions | Linear Algebra | Khan Academy
Surjective (onto) and injective (one-to-one) functions | Linear Algebra | Khan AcademyIntroduction to surjective and injective functionsWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/inverse_transf…
- 29 Oct 09, 5:45pm -
Proof: Invertibility implies a unique solution to f(x)=y | Linear Algebra | Khan Academy
Proof: Invertibility implies a unique solution to f(x)=y | Linear Algebra | Khan AcademyProof: Invertibility implies a unique solution to f(x)=y for all y in co-domain of f.Watch the next lesson: https://www.khanacademy.org/math/linear-algebra/…
- 29 Oct 09, 2:03am -
Introduction to the inverse of a function | Matrix transformations | Linear Algebra | Khan Academy
Introduction to the inverse of a function | Matrix transformations | Linear Algebra | Khan AcademyIntroduction to the inverse of a functionWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/inverse_tra…
- 28 Oct 09, 7:23pm -
Distributive property of matrix products | Matrix transformations | Linear Algebra | Khan Academy
Distributive property of matrix products | Matrix transformations | Linear Algebra | Khan AcademyShowing that matrix products are associativeWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/inverse_t…
- 28 Oct 09, 3:52pm -
Matrix product associativity | Matrix transformations | Linear Algebra | Khan Academy
Matrix product associativity | Matrix transformations | Linear Algebra | Khan AcademyShowing that matrix products are associativeWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/composition_of_transf…
- 28 Oct 09, 3:15pm -
Matrix product examples | Matrix transformations | Linear Algebra | Khan Academy
Matrix product examples | Matrix transformations | Linear Algebra | Khan AcademyExample of taking the product of two matricesWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/composition_of_transforma…
- 28 Oct 09, 2:01pm -
Compositions of linear transformations 2 | Matrix transformations | Linear Algebra | Khan Academy
Compositions of linear transformations 2 | Matrix transformations | Linear Algebra | Khan AcademyProviding the motivation for definition of matrix productsWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformat…
- 28 Oct 09, 1:16am -
Compositions of linear transformations 1 | Matrix transformations | Linear Algebra | Khan Academy
Compositions of linear transformations 1 | Matrix transformations | Linear Algebra | Khan AcademyIntroduction to compositions of Linear TransformationsWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations…
- 28 Oct 09, 1:16am -
Expressing a projection on to a line as a matrix vector prod | Linear Algebra | Khan Academy
Expressing a projection on to a line as a matrix vector prod | Linear Algebra | Khan AcademyExpressing a Projection on to a line as a Matrix Vector prodWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformation…
- 28 Oct 09, 1:16am -
Introduction to projections | Matrix transformations | Linear Algebra | Khan Academy
Introduction to projections | Matrix transformations | Linear Algebra | Khan AcademyDetermining the projection of a vector on s lineWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/lin_trans_examples…
- 28 Oct 09, 1:16am -
Unit vectors | Matrix transformations | Linear Algebra | Khan Academy
Unit vectors | Matrix transformations | Linear Algebra | Khan AcademyWhat unit vectors are and how to construct themWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/lin_trans_examples/v/introduction-…
- 28 Oct 09, 1:15am -
Rotation in R3 around the x-axis | Matrix transformations | Linear Algebra | Khan Academy
Rotation in R3 around the x-axis | Matrix transformations | Linear Algebra | Khan AcademyConstruction a rotation transformation in R3Watch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/lin_trans_example…
- 26 Oct 09, 5:17am -
Linear transformation examples: Rotations in R2 | Linear Algebra | Khan Academy
Linear transformation examples: Rotations in R2 | Linear Algebra | Khan AcademyLinear Transformation Examples: Rotations in R2Watch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/lin_trans_examples/v/rot…
- 23 Oct 09, 11:12pm -
Linear transformation examples: Scaling and reflections | Linear Algebra | Khan Academy
Linear transformation examples: Scaling and reflections | Linear Algebra | Khan AcademyCreating scaling and reflection transformation matrices (which are diagonal)Watch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_tra…
- 23 Oct 09, 11:12pm -
More on matrix addition and scalar multiplication | Linear Algebra | Khan Academy
More on matrix addition and scalar multiplication | Linear Algebra | Khan AcademyMore on Matrix Addition and Scalar MultiplicationWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/lin_trans_examples/v…
- 23 Oct 09, 5:06pm -
Sums and scalar multiples of linear transformations | Linear Algebra | Khan Academy
Sums and scalar multiples of linear transformations | Linear Algebra | Khan AcademySums and Scalar Multiples of Linear Transformations. Definitions of matrix addition and scalar multiplication.Watch the next lesson: https://www.khanacademy.org/…
- 23 Oct 09, 5:06pm -
Preimage and kernel example | Matrix transformations | Linear Algebra | Khan Academy
Preimage and kernel example | Matrix transformations | Linear Algebra | Khan AcademyExample involving the preimage of a set under a transformation. Definition of kernel of a transformation.Watch the next lesson: https://www.khanacademy.org/math…
- 22 Oct 09, 8:32pm -
Preimage of a set | Matrix transformations | Linear Algebra | Khan Academy
Preimage of a set | Matrix transformations | Linear Algebra | Khan AcademyDefinition of preimage of a setWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_transformations/linear_transformations/v/preimage-and-kernel-…
- 22 Oct 09, 7:46pm -
im(T): Image of a transformation | Matrix transformations | Linear Algebra | Khan Academy
im(T): Image of a transformation | Matrix transformations | Linear Algebra | Khan AcademyShowing that the image of a subspace under a transformation is also a subspace. Definition of the image of a Transformation.Watch the next lesson: https://…
- 22 Oct 09, 6:05pm -
Image of a subset under a transformation | Matrix transformations | Linear Algebra | Khan Academy
Image of a subset under a transformation | Matrix transformations | Linear Algebra | Khan AcademyExploring what happens to a subset of the domain under a transformationWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matri…
- 22 Oct 09, 6:05pm -
Linear transformations as matrix vector products | Linear Algebra | Khan Academy
Linear transformations as matrix vector products | Linear Algebra | Khan AcademyShowing how ANY linear transformation can be represented as a matrix vector productWatch the next lesson: https://www.khanacademy.org/math/linear-algebra/matrix_tra…
- 21 Oct 09, 1:12am -

Khan Academy - Linear Algebra 98-146

Linear Transformations
Linear TransformationsIntroduction to linear transformationsMore free lessons at: http://www.khanacademy.org/video?v=4PCktDZJH8EFrom:Khan AcademyViews:338528 610ratingsTime:13:53More inEducation
- 21 Oct 09, 1:12am -
Vector Transformations
Vector TransformationsIntroduction to the notion of vector transformationsMore free lessons at: http://www.khanacademy.org/video?v=gAbadNuQEjIFrom:Khan AcademyViews:141571 194ratingsTime:14:19More inEducation
- 21 Oct 09, 1:12am -
A more formal understanding of functions
A more formal understanding of functionsA more formal understanding of functionsMore free lessons at: http://www.khanacademy.org/video?v=BQMyeQOLvpgFrom:Khan AcademyViews:161955 203ratingsTime:16:02More inEducation
- 21 Oct 09, 1:12am -
Showing that the candidate basis does span C(A)
Showing that the candidate basis does span C(A)Showing that just the columns of A associated with the pivot columns of rref(A) do indeed span C(A).More free lessons at: http://www.khanacademy.org/video?v=CkQOCnLWPUAFrom:Khan AcademyViews:260…
- 19 Oct 09, 4:03am -
Showing relation between basis cols and pivot cols
Showing relation between basis cols and pivot colsShowing that linear independence of pivot columns implies linear independence of the corresponding columns in the original equationMore free lessons at: http://www.khanacademy.org/video?v=BfVjTOj…
- 19 Oct 09, 4:03am -
Dimension of the Column Space or Rank
Dimension of the Column Space or RankDimension of the Column Space or RankMore free lessons at: http://www.khanacademy.org/video?v=JUgrBkPteTgFrom:Khan AcademyViews:128210 342ratingsTime:12:48More inEducation
- 19 Oct 09, 4:03am -
Dimension of the Null Space or Nullity
Dimension of the Null Space or NullityDimension of the Null Space or NullityMore free lessons at: http://www.khanacademy.org/video?v=abYAUqs_n6IFrom:Khan AcademyViews:108239 247ratingsTime:13:59More inEducation
- 19 Oct 09, 4:03am -
Proof: Any subspace basis has same number of elements
Proof: Any subspace basis has same number of elementsProof: Any subspace basis has same number of elementsMore free lessons at: http://www.khanacademy.org/video?v=Zn2K8UIT8r4From:Khan AcademyViews:36621 63ratingsTime:21:35More inEd…
- 19 Oct 09, 4:02am -
Introduction to the Null Space of a Matrix
Introduction to the Null Space of a MatrixShowing that the Null Space of a Matrix is a valid SubspaceMore free lessons at: http://www.khanacademy.org/video?v=jCwRV1QL_XsFrom:Khan AcademyViews:196611 295ratingsTime:10:23More inEduca…
- 17 Oct 09, 6:33pm -
Visualizing a Column Space as a Plane in R3
Visualizing a Column Space as a Plane in R3Determining the planar equation for a column space in R3More free lessons at: http://www.khanacademy.org/video?v=EGNlXtjYABwFrom:Khan AcademyViews:46626 77ratingsTime:21:11More inEducation
- 17 Oct 09, 6:33pm -
Column Space of a Matrix
Column Space of a MatrixIntroduction to the column space of a matrixMore free lessons at: http://www.khanacademy.org/video?v=st6D5OdFV9MFrom:Khan AcademyViews:127762 226ratingsTime:10:40More inEducation
- 17 Oct 09, 6:33pm -
Null Space and Column Space Basis
Null Space and Column Space BasisFiguring out the null space and a basis of a column space for a matrixMore free lessons at: http://www.khanacademy.org/video?v=_uTAdf_AsfQFrom:Khan AcademyViews:206002 485ratingsTime:25:13More inEdu…
- 17 Oct 09, 6:33pm -
Null Space 2: Calculating the null space of a matrix
Null Space 2: Calculating the null space of a matrixCalculating the null space of a matrixMore free lessons at: http://www.khanacademy.org/video?v=qvyboGryeA8From:Khan AcademyViews:213424 473ratingsTime:13:08More inEducation
- 17 Oct 09, 6:33pm -
Null Space 3: Relation to Linear Independence
Null Space 3: Relation to Linear IndependenceUnderstanding how the null space of a matrix relates to the linear independence of its column vectorsMore free lessons at: http://www.khanacademy.org/video?v=vFctYRhK2M0From:Khan AcademyViews:8743…
- 17 Oct 09, 6:33pm -
Matrix Vector Products
Matrix Vector ProductsDefining and understanding what it means to take the product of a matrix and a vectorMore free lessons at: http://www.khanacademy.org/video?v=7Mo4S2wyMg4From:Khan AcademyViews:85658 122ratingsTime:21:10More in…
- 16 Oct 09, 4:48am -
Matrices: Reduced Row Echelon Form 3
Matrices: Reduced Row Echelon Form 3And another example of solving a system of linear equations by putting an augmented matrix into reduced row echelon formMore free lessons at: http://www.khanacademy.org/video?v=JVDrlTdzxiIFrom:Khan AcademyV…
- 15 Oct 09, 5:00pm -
Matrices: Reduced Row Echelon Form 1
Matrices: Reduced Row Echelon Form 1Solving a system of linear equations by putting an augmented matrix into reduced row echelon formMore free lessons at: http://www.khanacademy.org/video?v=L0CmbneYETsFrom:Khan AcademyViews:361811 588r…
- 15 Oct 09, 4:57pm -
Matrices: Reduced Row Echelon Form 2
Matrices: Reduced Row Echelon Form 2Another example of solving a system of linear equations by putting an augmented matrix into reduced row echelon formMore free lessons at: http://www.khanacademy.org/video?v=lP1DGtZ8WysFrom:Khan AcademyViews…
- 15 Oct 09, 4:57pm -
Dot and Cross Product Comparison/Intuition
Dot and Cross Product Comparison/IntuitionDot and Cross Product Comparison/IntuitionMore free lessons at: http://www.khanacademy.org/video?v=tdwFdzVqitoFrom:Khan AcademyViews:93244 254ratingsTime:19:14More inEducation
- 13 Oct 09, 11:23pm -
Proof: Relationship between cross product and sin of angle
Proof: Relationship between cross product and sin of angleProof: Relationship between the cross product and sin of angle between vectorsMore free lessons at: http://www.khanacademy.org/video?v=7MKA2QlKvHcFrom:Khan AcademyViews:62575 14…
- 13 Oct 09, 7:52pm -
Linear Algebra: Cross Product Introduction
Linear Algebra: Cross Product IntroductionIntroduction to the cross productMore free lessons at: http://www.khanacademy.org/video?v=pJzmiywagfYFrom:Khan AcademyViews:140596 186ratingsTime:15:47More inEducation
- 13 Oct 09, 7:51pm -
Defining a plane in R3 with a point and normal vector
Defining a plane in R3 with a point and normal vectorDetermining the equation for a plane in R3 using a point on the plane and a normal vectorMore free lessons at: http://www.khanacademy.org/video?v=UJxgcVaNTqYFrom:Khan AcademyViews:119073…
- 13 Oct 09, 7:51pm -
Defining the angle between vectors
Defining the angle between vectorsIntroducing the idea of an angle between two vectorsMore free lessons at: http://www.khanacademy.org/video?v=5AWob_z74KsFrom:Khan AcademyViews:139737 187ratingsTime:25:11More inEducation
- 13 Oct 09, 7:51pm -
Linear Algebra: Vector Triangle Inequality
Linear Algebra: Vector Triangle InequalityProving the triangle inequality for vectors in RnMore free lessons at: http://www.khanacademy.org/video?v=PsNidCBr5IIFrom:Khan AcademyViews:82143 175ratingsTime:18:53More inEducation
- 10 Oct 09, 5:53pm -
Proof of the Cauchy-Schwarz Inequality
Proof of the Cauchy-Schwarz InequalityProof of the Cauchy-Schwarz InequalityMore free lessons at: http://www.khanacademy.org/video?v=r2PogGDl8_UFrom:Khan AcademyViews:152964 250ratingsTime:16:56More inEducation
- 10 Oct 09, 3:08am -
Proving Vector Dot Product Properties
Proving Vector Dot Product PropertiesProving the "associative", "distributive" and "commutative" properties for vector dot products.More free lessons at: http://www.khanacademy.org/video?v=rVQ3G9epCjwFrom:Khan AcademyViews:114108 156ra…
- 10 Oct 09, 3:08am -
Vector Dot Product and Vector Length
Vector Dot Product and Vector LengthDefinitions of the vector dot product and vector lengthMore free lessons at: http://www.khanacademy.org/video?v=WNuIhXo39_kFrom:Khan AcademyViews:219114 230ratingsTime:09:10More inEducation
- 10 Oct 09, 3:07am -
Linear Algebra: Basis of a Subspace
Linear Algebra: Basis of a SubspaceUnderstanding the definition of a basis of a subspaceMore free lessons at: http://www.khanacademy.org/video?v=zntNi3-ybfQFrom:Khan AcademyViews:356415 714ratingsTime:19:00More inEducation
- 10 Oct 09, 3:07am -
Linear Subspaces
Linear SubspacesIntroduction to linear subspaces of RnMore free lessons at: http://www.khanacademy.org/video?v=pMFv6liWK4MFrom:Khan AcademyViews:379805 907ratingsTime:23:29More inEducation
- 10 Oct 09, 3:06am -
Span and Linear Independence Example
Span and Linear Independence ExampleDetermining whether 3 vectors are linearly independent and/or span R3More free lessons at: http://www.khanacademy.org/video?v=9kW6zFK5E5cFrom:Khan AcademyViews:231349 453ratingsTime:16:53More inE…
- 10 Oct 09, 3:06am -
More on linear independence
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- 10 Oct 09, 3:05am -
Linear Algebra: Introduction to Linear Independence
Linear Algebra: Introduction to Linear IndependenceLinear Algebra: Introduction to linear dependence and independenceMore free lessons at: http://www.khanacademy.org/video?v=CrV1xCWdY-gFrom:Khan AcademyViews:292834 549ratingsTime:15:…
- 10 Oct 09, 3:05am -
Linear Combinations and Span
Linear Combinations and SpanUnderstanding linear combinations and spans of vectorsMore free lessons at: http://www.khanacademy.org/video?v=Qm_OS-8COwUFrom:Khan AcademyViews:383645 770ratingsTime:20:35More inEducation
- 9 Oct 09, 12:48am -
Linear Algebra: Parametric Representations of Lines
Linear Algebra: Parametric Representations of LinesLinear Algebra: Parametric Representations of Lines in R2 and R3More free lessons at: http://www.khanacademy.org/video?v=hWhs2cIj7CwFrom:Khan AcademyViews:235714 378ratingsTime:24:47…
- 9 Oct 09, 12:47am -
Linear Algebra: Vector Examples
Linear Algebra: Vector ExamplesVisually understanding basic vector operationsMore free lessons at: http://www.khanacademy.org/video?v=r4bH66vYjssFrom:Khan AcademyViews:344086 500ratingsTime:25:34More inEducation
- 9 Oct 09, 12:46am -
Private video

- 9 Oct 09, 12:46am -
Solving 3 Equations with 3 Unknowns
Solving 3 Equations with 3 UnknownsSolving 3 equations with 3 unknowns (old video from July 2008)From:Khan AcademyViews:226232 587ratingsTime:15:26More inEducation
- 12 Apr 09, 4:11am -
3-variable linear equations (part 1)
3-variable linear equations (part 1)Visual intuition of a 3-variable linear equation.From:Khan AcademyViews:110726 178ratingsTime:08:02More inEducation
- 3 Jul 08, 12:10am -
Singular Matrices
Singular MatricesWhen and why you can't invert a matrix.More free lessons at: http://www.khanacademy.org/video?v=UqyN7-tRS00From:Khan AcademyViews:173490 291ratingsTime:14:27More inEducation
- 15 Jun 08, 5:43pm -
Matrices to solve a vector combination problem
Matrices to solve a vector combination problemUsing matrices to figure out if some combination of 2 vectors can create a 3rd vectorMore free lessons at: http://www.khanacademy.org/video?v=gsNgdVdAT1oFrom:Khan AcademyViews:162697 338rat…
- 14 Jun 08, 6:13pm -
Matrices to solve a system of equations
Matrices to solve a system of equationsUsing the inverse of a matrix to solve a system of equations.More free lessons at: http://www.khanacademy.org/video?v=AUqeb9Z3y3kFrom:Khan AcademyViews:555726 1005ratingsTime:16:33More inEduca…
- 14 Jun 08, 5:38pm -
Inverting Matrices (part 3)
Inverting Matrices (part 3)Using Gauss-Jordan elimination to invert a 3x3 matrix.More free lessons at: http://www.khanacademy.org/video?v=obts_JDS6_QFrom:Khan AcademyViews:232419 603ratingsTime:13:36More inEducation
- 8 Jun 08, 10:27pm -
Inverting matrices (part 2)
Inverting matrices (part 2)Inverting a 3x3 matrixMore free lessons at: http://www.khanacademy.org/video?v=S4n-tQZnU6oFrom:Khan AcademyViews:352669 829ratingsTime:16:45More inEducation
- 8 Jun 08, 8:05pm -
Inverse Matrix (part 1)
Inverse Matrix (part 1)Taking the inverse of a 2x2 matrixMore free lessons at: http://www.khanacademy.org/video?v=iUQR0enP7RQFrom:Khan AcademyViews:631442 1453ratingsTime:14:15More inEducation
- 8 Jun 08, 5:35pm -
Matrix multiplication (part 2)
Matrix multiplication (part 2)More on multiplying matrices.From:Khan AcademyViews:341129 954ratingsTime:14:37More inEducation
- 8 Jun 08, 3:33am -
Matrix multiplication (part 1)
Matrix multiplication (part 1)Multiplying two 2x2 matrices.From:Khan AcademyViews:619916 1533ratingsTime:13:41More inEducation
- 8 Jun 08, 3:33am -
Introduction to matrices
Introduction to matricesWhat a matrix is. How to add and subtract them.From:Khan AcademyViews:844790 2115ratingsTime:11:51More inEducation
- 8 Jun 08, 3:32am -