A square S has vertices (0,0), (2,0), (2,2) and (0,2). It is mapped onto a shape T by the transformation represented by the matrix A = [[2,-1],[1,2]]. Find the coordinates of the vertices of T.

Here is the solution to question 1: a) i) The vertices of square S are (0, 0), (2, 0), (2, 2), and (0, 2). (Drawing on a grid is not possible in this format). ii) Step 1: Identify the transformation matrix A from the given mapping. The mapping is x' \\ y' = 2x-y \\ x+2y . This corresponds to the matrix A = 2 & -1 \\ 1 & 2 . Step 2: Apply matrix A to each vertex of S. For P(0,0): P' = 2 & -1 \\ 1 & 2 0 \\ 0 = 0 \\ 0 For Q(2,0): Q' = 2 & -1 \\ 1 & 2 2 \\ 0 = 2(2) + (-1)(0) \\ 1(2) + 2(0) = 4 \\ 2 For R(2,2): R' = 2 & -1 \\ 1 & 2 2 \\ 2 = 2(2) + (-1)(2) \\ 1(2) + 2(2) = 4 - 2 \\ 2 + 4 = 2 \\ 6 For S(0,2): S' = 2 & -1 \\ 1 & 2 0 \\ 2 = 2(0) + (-1)(2) \\ 1(0) + 2(2) = -2 \\ 4 The coordinates of the image T are P'(0,0), Q'(4,2), R'(2,6), S'(-2,4). (Drawing on a grid is not possible in this format). iii) Step 1: Identify the transformation matrix B. The matrix is B = 2 & 1 \\ -1 & 2 . Step 2: Apply matrix B to each vertex of S. For P(0,0): P'' = 2 & 1 \\ -1 & 2 0 \\ 0 = 0 \\ 0 For Q(2,0): Q'' = 2 & 1 \\ -1 & 2 2 \\ 0 = 2(2) + 1(0) \\ -1(2) + 2(0) = 4 \\ -2 For R(2,2): R'' = 2 & 1 \\ -1 & 2 2 \\ 2 = 2(2) + 1(2) \\ -1(2) + 2(2) = 4 + 2 \\ -2 + 4 = 6 \\ 2 For S(0,2): S'' = 2 & 1 \\ -1 & 2 0 \\ 2 = 2(0) + 1(2) \\ -1(0) + 2(2) = 2 \\ 4 The coordinates of the image U are P''(0,0), Q''(4,-2), R''(6,2), S''(2,4). (Drawing on a grid is not possible in this format). b) i) Step 1: Find the product AB. AB = 2 & -1 \\ 1 & 2 2 & 1 \\ -1 & 2 AB = (2)(2) + (-1)(-1) & (2)(1) + (-1)(2) \\ (1)(2) + (2)(-1) & (1)(1) + (2)(2) AB = 4 + 1 & 2 - 2 \\ 2 - 2 & 1 + 4 AB = 5 & 0 \\ 0 & 5 Step 2: Apply the matrix AB to each vertex of S to find the image V. For P(0,0): P''' = 5 & 0 \\ 0 & 5 0 \\ 0 = 0 \\ 0 For Q(2,0): Q''' = 5 & 0 \\ 0 & 5 2 \\ 0 = 10