1. [2 points each] Let A be an m x n matrix. Define Ker(A) := {X E R | Ax = 0} and Im(A) :=
1. [2 points each] Let A be an m x n matrix. Define Ker(A) := {X E R" | Ax = 0} and Im(A) := {y E Rm | there is a x E R" such that Ax = y }. (a) Show that Ker(A) is a subspace of R" and Im (A) is a subspace of Rm. (b) Show that dim Ker( A) = dim null(A), and that dim Im(A) = rank(A). (c) Show that dim Ker( A) + dim ImA = n
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