1. Suppose T : R² → R³ is a linear transformation defined by " (E) - T T 9 -3 Find the matrix of T with respect to the standard bases E and Ez for R? and R3 respectively.

Find the standard matrix of the linear transformation T: R² → R³ given by T(x₁, x2) = (2x₁ - 3x2, x₁ x1 1 1 [23 -3 -2 0 1 0 0 2 1 0 -2 0 0 2 -3 1 -2 2 -3 H 1 -2 1 -3 Го 2x2, x₂).

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

Suppose that T:R²→ R³ is a linear transformation and T(1,-3)=(4,–3, 1) and T(1,4)=(1,3, 1) and let B={(1,–3),(1,4)}. a) Find the matrix representation of Tin terms of B and S3 [T] where S3 is the standard basis for R'. b) Find the change of basis matrix [C, to convert from the standard basis in R? to B. c) Find [T] = [TCE, ->

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|

Suppose T: R³→R² is a linear transformation. Let U and V be the vectors given below, and suppose that T(U) and T(V) are as given. Find T(−U+2V). 2 U = 1 3 4 V = 3 4 0 T(-U+2V) = 0 0 -6 [:] 6 T(U)= -9 [3] 9 T(V) =

3. Consider the linear transformation T: R² → R² that projects orthogonally onto the y-axis. (a) As a review, determine T(₁) and T(2) geometrically and use your answers to write the matrix for T. (b) Show how to repeat the process we used in the earlier problems to find the matrix for T. Note: you should get the same matrix, of course.

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