Let T be the function T: R³ R2 given by + (1)-[*] (ED) T (a) Show that T is a linear transformation. (b) Determine the standard matrix for T. (c) Find a basis for ker(T).

Let the linear transformation T: R³ → R² be defined by T(1₁, 12, 13) = (₁ + I2, I2+I3) (a) Find the standard matrix of T. (b) Determine whether T is (i) one-to-one and (ii) onto. Justify your answer.

Define a linear operator (transformation) on R2R2 by performing the following operations in order: (i) a rotation by an angle of π/4π/4 clockwise, (ii) an orthogonal projection onto the vector u=(1,1)u=(1,1), and (iii) a compression in the x−x−direction by a factor of 22. Compute the standard matrix of this operator.

Let a and B be the standard bases for R" and R" respectively. For each of the following linear transformations T; : R² → R", compute the matrix [T;%- 2a b () (a) T1 : R² → R³, T За + 46 a () - (*:) (C) (C) (2а + 3b —. a +b (b) Т, : R3 — R?, T, b (c) T3 : R³ → R', T3 = (2a + b – 3c). (d) T4 : R" → R", T4