Khan Academy
Khan Academy: Linear Algebra: (Correction) Scalar Multiplication of Row
A brief video correcting a notational mistake that was made in the previous video on how scalar multiplication affects the determinant. The formula in the previous video shows the entries being multiplied by the sub-matrices, when really...
Khan Academy
Khan Academy: Linear Algebra: Matrix Product Examples
Video shows example of multiplying matrices. Includes explanation of the dimensions needed in order for matrices to be able to be multiplied. Connects to the composition of linear transformations. This video also appears in the strand...
Khan Academy
Khan Academy: Linear Algebra: Determinant of a Transpose
Video demonstrating that the determinant of a matrix is equivalent to the determinant of the transpose of a matrix. Proof by induction is used to prove this property. [14:10]
Khan Academy
Khan Academy: Linear Algebra: Another Example of a Projection Matrix
A video lesson working another example for finding the projection of an arbitrary vector onto a specific subspace in R3. The basis of the subspace is a 3 x 2 matrix. Uses an alternate method for find the transformation matrix that...
Khan Academy
Khan Academy: Linear Algebra: Simplifying Conditions for Invertibility
A video lesson explaining that a transformation matrix is invertible if and only if it is a square identity matrix in reduced row echelon form. This video is also found in the strand Algebra: Matrices. [6:37]
Khan Academy
Khan Academy: Linear Algebra: Formula for 2 X 2 Inverse
A video lesson in which a formula is derived for finding the inverse of a 2 x 2 matrix. Addresses that a 2 x 2 matrix is only invertible if the determinant is not equal to 0. Defines the determinant and shows two examples of calculating...
Khan Academy
Khan Academy: Linear Algebra: Linear Transformations: Scaling and Reflections
Video first reviews how to find the transformation matrix using the identity matrix. Shows how to write a transformation matrix that reflects over the y-axis and scales the y-coordinates by a factor of 2. [15:13]
Khan Academy
Khan Academy: Linear Algebra: Compositions of Linear Transformations 2
Video shows that the transformation matrix of the composition of linear functions is the product of the transformation matrices of each linear transformation in the composition. [16:30]
Khan Academy
Khan Academy: Showing That an Eigenbasis Makes for Good Coordinate Systems
Video creating a basis out of n linearly independent eigenvectors. Reviews transformations and change of basis matrices. Shows that when the eigenbasis is used as an alternate coordinate system, the transformation is a diagonal matrix...
Khan Academy
Khan Academy: Determining Whether a Transformation Is Onto
Shows how to determine if a linear transformation is onto by showing that the columns of the transformation matrix span the co-domain.
Khan Academy
Khan Academy: Change of Basis: Alternate Basis Transformation Matrix Part 2
Showing 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.
Khan Academy
Khan Academy: More Determinant Depth: Determinant as Scaling Factor
Video demonstrating that the area of a parallelogram that is the image of a rectangle under a transformation is equivalent to the absolute value of the determinant of the matrix whose column vectors generate the parallelogram.
Khan Academy
Khan Academy: Using Orthogonal Change of Basis Matrix
Video defining an orthogonal matrix as a square matrix C whose columns form an orthonormal set.
Khan Academy
Khan Academy: More Determinant Depth: Duplicate Row Determinant
A video lesson proving that if a matrix has duplicate rows then its determinant will be 0.
Khan Academy
Khan Academy: Finding Inverses of Matrices: Nxn Determinant
Defining the determinant for nxn matrices. An exampled of a 4x4 determinant.
Khan Academy
Khan Academy: Linear Transformation Examples
Expressing a Projection on to a line as a Matrix Vector prod
Khan Academy
Khan Academy: Changing Coordinate Systems to Help Find a Transformation Matrix
Changing our coordinate system to find the transformation matrix with respect to standard coordinates
Khan Academy
Khan Academy: Change of Basis: Alternate Basis Transformation Matrix Example
Example of finding the transformation matrix for an alternate basis.
Khan Academy
Khan Academy: Invertible Change of Basis Matrix
Using an invertible change of basis matrix to go between different coordinate systems.
Khan Academy
Khan Academy: Finding Inverses of Matrices: Determinants Along Other Rows/cols
Finding the determinant by going along other rows or columns
Khan Academy
Khan Academy: Transformation Matrix With Respect to a Basis
Finding the transformation matrix with respect to a non-standard basis.
Khan Academy
Khan Academy: More Determinant Depth: Simpler 4 X4 Determinant
This video, shows how to calculate a 4x4 determinant by putting in in upper triangular form first.
Khan Academy
Khan Academy: Linear Algebra: Inverting Matrices (Part 2)
Video example of finding the inverse of a 3 by 3 matrix. Includes using expansion of minors to determine the determinant and finding the matrix of cofactors and adjugate or adjoint matrix. This video also appears in the strand Algebra:...
Khan Academy
Khan Academy: Linear Algebra: Inverting Matrices (Part 2)
Video example of finding the inverse of a 3 by 3 matrix. Includes using expansion of minors to determine the determinant and finding the matrix of cofactors and adjugate or adjoint matrix. This video also appears in the strand Algebra:...