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1. Solve the following systems of linear equations. (i) 4x1 − 3x2 + 2x3 − x4 = 8 3x1 − 2x2 + x3 − 3x4 = 7 2x1 − x2 − 5x4 = 6 5x1 − 3x2 + x3 − 8x4 = 1. (ii) 2x + y + 4z + 8t = −1 x + 3y − 6z + 2t = 3 3x − 2y + 2z − 2t = 8 2x − y + 2z = 4. 2. Find the rank and the determinant of the matrix A =   λ 1 1 2 1 + λ 2 λ 2λ − 1 λ   for all λ ∈ R. Find all λ such that the system Ax =  1 2 2T has a unique solution x ∈ R 3 . 3. Find the eigenvalues and diagonalize the matrix A = 3 2 −4 −3 ! Compute A2017 . 4. Write the following functions in a matrix form and find the range of each of these functions. Specify which of these functions is injective, surjective, and bijective. (a) f(x1, x2, x3) = (x2, x1, x3) T . (b) g(x1, x2, x3) = 3x 2 1 + 2x 2 2 + 4x1x2 + 2x 2 3 − x2x3 − 3x1x3. (c) g(x1, x2, x3) = 2x 2 1 − 3x 2 2 − 2x1x2 + 4x 2 3 − 3x2x3 + 4x1x3. 5. For each of the following subsets of Euclidean spaces, say whether it is closed, open, compact. (a) A1 = R \ Q = {x ∈ R : x is irrational}. (b) A2 = {(x, y, z) ∈ R 3 : x ≥ 0, y ≥ 0, z ≥ 0, ex + e y + e z ≤ 5}. (c) A3 = {(x, y) ∈ R 2 : x 6= 0, y = 1/x}. (d) A4 = {(x, y) ∈ R 2 : 0 < x6 + y 6 < 9}

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