Let T : M2x2 (R) → M2x2 (R) be the linear transformation defined by "(: :)-( ) "). 2x z y - z+ w T 2w 2w – z Is T diagonalizable? If so, find B so that [T], is diagonal.

Let Tv = Av represent the linear transformation T: R2 → R3 where 1 2 A = -2 4 -2 2]

Letf: R² → R be defined by f((x, y)) = -6x - 8y. Isf a linear transformation? a. f((x₁, y₁) + (x2, y₂)) = b. f(c(x, y)) = f((x₁, y₁)) + f((x₂, y₂)) = + Does f(x,y1) + (x₂, y2)) =f((x₁, y₁)) + f((x2, y2)) for all (x₁, y₁), (x₂, y2) E R²? choose (Enter x₁ as x1, etc.) c(f((x, y))) = Does f(c(x, y)) = c(f((x, y))) for all c ER and all (x, y) E R²? choose c. Isf a linear transformation? choose + + +

Prove that T(x1,x2,?3) = [x1,x2] is a linear transformation.

2. Let T : R² → R² be a linear transformation such that: 1 -2 (i) T maps vector u = into the vector a = [ 5 3 (ii) T maps vector v = [³] into the vector b = [ Use the fact that T is a linear transformation to find T(3u +2v). -4

17. Let T R² R² be a linear transformation that maps [1] [3] 3 Tis linear to find the images under T of 3u, 2v, and 3u + 2v. u= into and maps v = into Use the fact that

Let T: R² →R² be a linear transformation such that T (×₁,×₂) = (x₁ +×₂, 6x₁ +5x2). Find x such that T(x) = (1,12). X =