Khan Academy: Linear Algebra: Showing That a Transpose X a Is Invertible

Showing that (transpose of A)(A) is invertible if A has linearly independent columns in this short video.

A video lesson that gives an in-depth, algebraic proof showing that the inverse of a linear transformation is also linear. [21:25]

Video explaining that the transpose of a matrix is created by switching rows and columns. The idea is shown abstractly and specifically with 2x2 matrix and 4x3 matrix. Also shows that the transpose of a transpose is the original matrix....

Video first reviews how to find the basis for the column space of a matrix using the reduced row echelon form. Proves that the basis for the column space using the reduced row echelon form method spans the column space. Uses this and the...

Video first reviews what it means for a function to be invertible. Shows that if a function is invertible then there is a unique solution to f(x) = y. Also shows that if there is a unique solution to the equation f(x) = y for all y then...

Video shows that the distributive property is true for multiplying matrices. [9:52]

Video reviews what the null space of a matrix is and how to calculate it, then relates it to linear independence. Shows that if the vectors are linearly independent then the null space is the zero vector. Also shows that if the null...

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]

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...

Video starts with reviewing matrix notation. Defines multiplying a matrix by a vector and shows examples. Shows how the matrix-vector products can be interpreted as the dot product of column vectors with a matrix or as a weighted linear...