2. Suppose that T R² R² is a linear transformation with the : properties that (3)-B] (7)-[(4) D = 1 Find a matrix B so that, for every x E R², T(x) = Bx. (a) T (b) T

Let 9 2 A 0 7 and B = Find the matrix C of the linear transformation T(x) = B(A(x)). C =

Find the matrix of the linear transformation T from R2 - R² where T([1, 0]) = [1, –2] and T([2, 1]) = [2, 3] %3D (Note: matrix A = [T([1, 0]), T([0, 1])] 1 -2 (a) 1 (b) 1 (c) -2 (d) -2 3 7 3 Ob a C

Find the matrix M of the linear transformation T: R² → R² given by T ([21]) - [-5²₁ + (-7) ²1]. = -5x1-x₂ M:

Suppose T is a linear transformation on R² (i.e. T(x, y) = (ax + by, cx + dy) where a, b, c, d are real numbers and, interpreting T as a matrix, det(T) # 0. Let P be the parallelogram which is the image of the square D = = [0, 1] × [0, 1] (i.e. T(D) =P). Then the area of P is given by O lad - bcl ad bc abcd 1 |ad - bc| O labcd|

Let T : R³ → R³ be the linear transformation that does the following things, in this order, to an input vector x = [x y z]¹: (i) Interchanges the second and third coordinates of . (ii) Multiplies the first coordinate of the resulting vector by 2. (iii) Replaces the second coordinate of the resulting vector with a 0. (iv) Multiplies the resulting vector by the following matrix: 0 0 0 0 You don't have to show that T is linear. (a) The description of T given above is purely algebraic, in that it explicitly describes how to take 7 = [x y z] and write down T() in coordinates. Give a geometric description of what each of the four "steps" of applying T actually does to a vector. (Your Week 9 tutorials may help in describing what the last step does.) (b) Find the standard matrix AT of T. (c) Find a spanning set for null(AT), and describe what null(AT) is geometrically (i.e., describe it geomet- rically as a subset of R³) (d) Find a spanning set for im(AT), and describe what im(AȚ) is…

Suppose T: R³-R² is a linear transformation. Let U, V and W be the vectors given below, and suppose that T(U) and T(V) are as given. Find T(W). -3 3 2 3 ਪ = -3 -1 0 T(W) = 0 0 V = -3 W = | 3 - 1 T(I) = 0 T(V) = 0 -3

Consider a linear transformation To R₂ [x] →> R²³² whose matrix relative to the bases B = {X², X, I is MT (B, C) = 0 2 and C= {(0), (6), (8)} 2 1) Find a formula for T. That is, what is T(a+bx+cx²) ?

7. Find the standard matrix for the linear transformation T. T:R²→R², T maps e¡ to e1 +4e2 but leaves e2 unchanged.

4. Find the standard matrix of the linear transformation T: R² → R² that reflects points across the line y = -1. [A 9 [8]] C D