Problem 3

(1 point) Suppose f(en) = an for n = 1, 2 and f is a linear transformation. e2 -2 -3 -2 -1 3_92 -2 -1 12 Domain Codomain a. Find a formula for f. Your answer should be a coordinate vector with the variables a and y in its components. f(I, y) = b. Find the matrix for the linear transformation f (relative to the standard basis in the domain and codomain). That is, find the matrix A such that f(x) = Ax. For instance, enter [ [1,2], [3,4] ] for the matrix . A = c. Find the kernel of f. Enter your answer as a vector with constant entries, a vector with the variables a or y (or both) in its components (using a minimum number of variables), or enter R^2 for IR2 ker ( f) = d. Find the image of f. Enter your answer as a vector with constant entries, a vector with the variables a or y (or both) in its components (using a minimum number of variables), or enter R^2 for IR2. im(f) = e. The linear transformation f is (select all that apply): O A. injective (one-to-one OB. bijective (an isomorphism) C. surjective (onto) D. none of these a. The domain of f is . For instance, enter R*5 for IR.5. b. The codomain of f is For instance, enter R^5 for IR 5 . c. Select all of the vectors that are in the kernel of f. You should be able to justify your answers (there may be more than one correct answer) JA. OB. c. OD. JE. OF. d. Select all of the vectors that are in the image of f. You should be able to justify your answers (there may be more than one correct answer DA. OB. C. OD. DE. JF.