Suppose that T is a linear transformation from R2 to the space of 2 × 2 matrices which satisfies: T((1, 1)) = [} 1 3 T((3,-2)) = What is T(2,4)? HINT: Note that - 2 (2,4) = 10(1,1) - (3,-2)

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

Consider the matrix A = a1 a12 a13 a14 24, show that the product L(x)=Ai is a linear transformation, where žER“.

Let = A = - = The cross product of two vectors in R³ is defined by a1 a2 az X b₁ b₂ b3 = 0 E] 7 Find the matrix A of the linear transformation from R³ to R³ given by T(x) = V × ã. [a₂b3-a3b₂ a3b₁a₁b3 [a₁b₂-a₂b₁]

Find the standard matrix for the linear transformation T(1, 9) — (2а — Зу, т — у, у — 4г) -3 1 -1 1 1 -4 2 1 -4 -3 -1 1 -3 -1 2 1 1 -3 -1 2.

Please answer question 2

Apply Euler's method twice to approximate the solution to the initial value problem on the [0./1/1 interval 0, first with step size h = 0.25, then with step size h = 0.1. Compare 1 the three-decimal-place values of the two approximations at x = the actual solution. y' = y + 3x - 6, y(0) = 2, y(x) = 3 − 3x - ex y (127) of with the value of y

2. Let (r x s) denote an r x s matrix. Find the size of those matrix products that are defined. a) (2 x 3)(3 x 4) b) (1 × 2)(3 × 1) c) (4 × 4)(3 × 3) d) (4 x 1)(1 x 2) e) (5 x 2)(2 x 3)

Please two only

For the matrices [1 3 B = 2 1 5 0 2 A = -5 4 and 1 8 Compute 2A + B'.