13. (a) Let A R2. Prove that (Write A(ax+By)=aAx+ BAy for all a, R, x, y...

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13. (a) Let A R2. Prove that (Write A(ax+By)=aAx+ BAy for all a, R, x, y R. A-[]-[]-[] ] x = [ 2 ] y = [ 22 ] X= x2 = all a12 921 922 and write out A(ax+By) explicitly.) (b) Now repeat part 13(a) for A & R"x". The proof is not difficult when one uses summation notation. For example, if the (i, j)-entry of A is denoted a;;, then (Ax); = a;jxj. j=1 Write (A(ax+By)); and (aAx+ BAy), in summation notation, and show that the first can be rewritten as the second using the elementary properties of arith- metic. 13. (a) Let A R2. Prove that (Write A(ax+By)=aAx+ BAy for all a, R, x, y R. A-[]-[]-[] ] x = [ 2 ] y = [ 22 ] X= x2 = all a12 921 922 and write out A(ax+By) explicitly.) (b) Now repeat part 13(a) for A & R"x". The proof is not difficult when one uses summation notation. For example, if the (i, j)-entry of A is denoted a;;, then (Ax); = a;jxj. j=1 Write (A(ax+By)); and (aAx+ BAy), in summation notation, and show that the first can be rewritten as the second using the elementary properties of arith- metic.