Question

-3 4 7. Prove or disprove, A: R3 R3 is bijective (1-1 and onto), where the standard matrix for A is A = -2 1 -1 3 8. Let A: R2 R2 be the linear transformation that that stretches the a-axis by a factor of 3, and the y-axis by a factor of 4. Find the standard matrix for A. 127

both questions

Answer 1

A: R^3 rightarrow R^3 where the standard matrix is, A = [1 2 -1 -3 1 5 4 -2 3] det(A) = 1(3 + 10) + 3(6 -2) + 4(10 + 1) = 13 + 12 + 44 = 69 notequalto 0 Since det(A) notequalto 0 and A is a map from R^3 to it self so it is bijective Because det(A) notequalto 0 implies A^-1 exist Now Ax = 0 implies x = A^-1 = 0 implies Nullity of A = {0} implies A is 1 -1 Also for each y elementof R^3 A(A^-1 y) = (AA^-1)y = IY = y implies y has a pre-image under A for all y elementof R^3 implies A is onto Hence A is a bijection So true. A: R^2 rightarrow R^2 be the linear transformation that stretches x-axis by a factor of 3 implies A(1 0) = 3(1 0) Also it stretches y-axis by a factor 4 implies A(0 1) =

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