3Blue1Brown
Matrix Multiplication as Composition | Essence of Linear Algebra, Chapter 4
Take a different view at matrix multiplication. Based on vector transformations, the video presents matrix multiplication as the result of the composition of multiple linear transformations. The presentation explains the numerical...
3Blue1Brown
Linear Transformations and Matrices | Essence of Linear Algebra, Chapter 3
Are all transformations of the plane linear transformations? The video provides examples of transformations of the plane and provides the definition of linear transformations. The resource begins the discussion of using matrix...
Khan Academy
Khan Academy: Linear Algebra: Determinant When Row Multiplied by Scalar
An instructional video explaining what happens to the determinant when one row in a matrix is multiplied by a scalar. Proves that the new determinant will simply be the scalar times the original determinant. Also addresses the case when...
Khan Academy
Khan Academy: Linear Algebra: Eigenvectors and Eigenspaces for a 3x3 Matrix
Video finding the eigenspace for the two eigenvalues for the specific 3x3 matrix from the previous video. Part of the process of solving requires finding a null space by getting a matrix into reduced row echelon form. The resulting two...
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: Linear Algebra: Why Eigenbases Make 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: Linear Algebra: Showing That a Transpose X a Is Invertible
A video lesson proving that the product of a matrix and its transpose is invertible if all of the columns of the originial matrix are linearly independent. [12:34]
Khan Academy
Khan Academy: Linear Algebra: 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. Includes...
Khan Academy
Khan Academy: Linear Algebra: Duplicate Row Determinant
A video lesson proving that if a matrix has duplicate rows then its determinant will be 0. It follows then that such a matrix will not be invertible. [8:19]
Khan Academy
Khan Academy: Linear Algebra: Change Bases to Help Find a Transformation Matrix
Video showing how to change coordinate systems to make finding the transformation matrix with respect to the standard basis easier. First, a specific linear transformation is defined and the regular method for constructing the...
Khan Academy
Khan Academy: Linear Algebra: Determinant After Row Operations
A video lesson proving that the determinant of a matrix does not change even after performing row operations on the matrix. [10:25]
Khan Academy
Khan Academy: Linear Algebra: Alternate Basis Transformation Matrix Example
Video reviewing the relationship between transformation matrices with respect to different bases. Addresses manipulating the equation to write it in terms of the transformation matrix with respect to the standard basis. The relationship...
Khan Academy
Khan Academy: Linear Algebra: Linear Transformation Examples: Rotations in R2
Video first shows visually that a rotation is a linear transformation. Shows how to construct a transformation matrix that will rotate a vector through a given angle. Shows both the general rotation matrix and an example of a rotation...
Khan Academy
Khan Academy: Linear Algebra: Invertible Change of Basis Matrix
A video lesson explaining how to convert between traditional Cartesian coordinates and coordinates with respect to a given basis by using an invertible change of basis matrix. Includes two concrete examples in R2. [13:34]
Khan Academy
Khan Academy: Linear Algebra: Showing That Inverses Are Linear
A video lesson that gives an in-depth, algebraic proof showing that the inverse of a linear transformation is also linear. [21:25]
Khan Academy
Khan Academy: Linear Algebra: Rule of Sarrus of Determinants
A video lesson presenting the Rule of Sarrus for finding the determinants of 3 x 3 matrices. The rule is derived algebraically and then used in a concrete example to find the determinant of a given 3 x 3 matrix. [7:18]
Khan Academy
Khan Academy: Linear Algebra: Determinant and Area of a Parallelogram
Video demonstrating that the area of a parallelogram created by graphing the column vectors of a 2x2 matrix is the equivalent to the absolute value of the determinant of the matrix. Pythagorean theorem, in conjunction with calculating a...
Khan Academy
Khan Academy: Linear Algebra: Determinants Along Other Rows/cols
A video lesson describing how the determinants for larger matrices can be found using cofactor expansion on any row or column. Illustrates that it is helpful to choose rows or columns with zeros in order to simplify the computations....
Khan Academy
Khan Academy: Linear Algebra: Transformation Matrix With Respect to a Basis
Video discussing transformation matrices and how a vector is able to be mapped to the same resulting vector by using a transformation matrix with respect to the standard basis as well as one with respect to a different basis. Reviews how...
Khan Academy
Khan Academy: Linear Algebra: Simpler 4x4 Determinant
Video demonstrating how to find the determinant of a 4x4 matrix by using row operations and row replacements to get the matrix into upper triangular form. [9:13]
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 by 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: Singular Matrices
Video explains why a matrix is singular if its determinant is 0. Illustrates that when a matrix equation is built with a singular matrix, then there are no solutions because the system it represents is that of two parallel lines. Shows...
Khan Academy
Khan Academy: Linear Algebra: Determinant When Row Is Added
A video lesson explaining the determinant rule for the very special case when three matrices are identical except for one row, and that row of the third matrix is the sum of those rows in the other two matrices. Proves that the...
Khan Academy
Khan Academy: Linear Algebra: Upper Triangular Determinant
A video lesson proving that the determinant of an upper triangular matrix is the product of the all of the entries along the main diagonal. Includes a concrete example. [8:07]