Suppose that T : R' → R' is a linear transformation whose standard matrix has 2 pivots. Is it possible to find two different vectors u, v such that T(u) =T(v)? That is, is (i) T one-to-one? Answer and explain briefly how you know. (ii) Given any vector b in R2, is it possible to find x such that T(x) = b? That is, does T map onto R?. Answer and explain briefly how you know.

Find a preimage of the vector 4 V= under the linear transformation T given by the matrix -3] = [44] -4 14 That is find the vector u such that T(u) = v. A U₁ ū₂

5. Consider a linear transformation T: R2 → R2 which reflects a vector about the line y=-x, and dilates the reflected vector by a factor of 2. Find the standard matrix for the linear transformation.

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…