Problem 1 Find a basis for the kernel and the image of the following maps. Also find the dimension of the kernel and image. (a) Matrix A (defined below) be a linear map A: R4 → R³ (b) Let G R³ R³ be the map 1 2 3 1 A = 1 3 5 -2 3 8 13 -3 G(x, y, z) = (x + 2y − z, y + z, x + y − 2z)

2. Let A = be an orthogonal 2 × 2 matrix. cos(0) sin(0 (a) Explain why the vector can be written as for some value of 0. a Use your answer to part (a) to find all 2×2 orthogonal matrices A = |C (b) (Hint: What can the second column of A look like?) (c) det(A) = 1? What kind of linear transformation do they correspond to? ! Given your answer to part (b), what are the 2 × 2 orthogonal matrices with

Suppose T is a linear transformation on R² (i.e. T(x, y) = (ax + by, cx + dy) where a, b, c, d are real numbers and, interpreting T as a matrix, det(T) # 0. Let P be the parallelogram which is the image of the square D = = [0, 1] × [0, 1] (i.e. T(D) =P). Then the area of P is given by O lad - bcl ad bc abcd 1 |ad - bc| O labcd|

Suppose we have a linear transformation T where T: P3 → P₂. What are the dimensions of the matrix A for T? (a) A is a 3x3 matrix. (b) A is a 4x3 matrix. (c) A is a 3x2 matrix. (d) We can't determine the size of A without knowing the specific bases on P3 and P₂. (e) A is a 2x3 matrix.