Khan Academy: Linear Algebra: Matrix Condition for One to One Trans

Video first reviews what the null space is and how to solve for it using reduced row echelon form. Shows that any solution to the inhomogeneous system Ax = b will be the sum of a particular solution and a homogeneous solution. Extends this to show that for the transformation to be one-to-one the null space must only contain the zero vector which means the column vectors of the transformation matrix are linearly independent and a basis for the column space. Shows that if a linear transformation is one-to-one, then the rank of the transformation matrix is equal to its number of columns. This video also appears in the strand Algebra: Matrices. [19:59]