MAT137Y1 Lecture Notes - Lecture 28: Linear Map, Invertible Matrix
MAT137Y1 verified notes
28/39View all
Document Summary
Big theorm: the big theorem only applies when m = n. Theorem 1: suppose t : r m r n a linear transformation and a is an n m matrix with t(x) = ax, and a = [a1am]. If m > n, none of the following hold. If m n, the following are equivalent. (i. e. they are all true or all false. ) The columns of a are linearly independent (i. e. {a1, , am} is a linearly independent set. ) Ax = b has at most one solution for all b r n . Theorem 2: suppose t : r m r n a linear transformation and a is an n m matrix with t(x) = ax, and a = [a1am]. If m < n, none of the following hold. {a1, , am} spans r n (i. e. span{a1, , am} = r n . ) Ax = b has at least one solution for all b r n .
