Question: all 412 ain 021 ( 22 a2n Let A = E Mmxn(R). Denote the vectors associated with the rows . . . aml am2 .

all 412 ain 021 ( 22 a2n Let A = E Mmxn(R). Denote the vectors associated with the rows . . . aml am2 . . . akl ak2 of A be al, ..., am. That is, ak = ( R" for all k = 1, .... m. We will walk through the . . . akn] steps to finally prove that (Row(A)) = Null(A) and (Col(A)) + = Null(AT). al . C (a) Let = = ( R" and denote ZA = Compute all entries of the vector AT . . . am . I and of the vector TA in terms of entries of A and entries of . What is the relation between Ar and CA? v1 U2 (b) Let 7 = (Row(A)). Explain in no more than three sentences why ak . 7 = 0 . . . Un for all k = 1, ..., m. From there, explain why ve Null(A). (c) Let we Null(A) and Z E Row(A). Prove that w . Z = 0. (d) Prove that ( Row(A))! = Null(A) and (Col(A) ) + = Null(AT)

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