Question: 1. [2 points each] Let A be an m x n matrix. Define Ker(A) := { x ( R | Ax = 0} and Im(A)

1. [2 points each] Let A be an m x n matrix. Define Ker(A) := { x ( R" | Ax = 0} and Im(A) := {y E R" | there is a x ( 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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