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Linear Algebra I, Practice Final exam  1 −1 1  −1 1 1   1. (14 points) All parts of this problem refer to the following matrix: A =   1 −1 −1 . −1 1 −1 

(a) (2 pts) Find rref(A), the reduced row echelon form of A. (b) (1 pt) Find the rank of A. (c) (3 pts) Find dim N (A), the null space of A, and find a basis of N (A). (d) (3 pts) Find dim C(A), the column space of A, and find an orthonormal basis of C(A).   1  1   (e) (2 pts) Check whether the vector u =   1  is in C(A). 1   3  −1   (f) (3 pts) Solve the system Ax = b where b =   1 . −3 

 1 −1 0 2 −1 . 2. (15 points) All parts of this problem refer to the following matrix A =  −1 2 −1 1 (a) (2 pts) Find det A using the cofactor method. (b) (1 pt) Find det A using Gaussian Elimination. (c) (4 pts) Find A−1 using the cofactor method. (d) (3 pts) Find A−1 using Gauss-Jordan Elimination. 

 0 (e) (3 pts) Consider the system of equations Ax = b where b =  −2 . Use 4 Cramer’s rule to find only x1 , the first variable. (f) (2 pts) Solve the system Ax = b in part (e) (you have to find x1 , x2 , x3 ) any method you like.  1 0 3. (8 points) All parts of this problem refer to the following matrix A =  0 1 1 0 (a) (3 pts) Find χA (λ), the characteristic polynomial of A. (b) (5 pts) Find the eigenvalues of A and their corresponding eigenvectors.

using  1 0 . 1

2 

 −1 2 1 4. (12 points) All parts of this problem refer to the following matrix A =  −1 2 1 . −2 2 2 (a) (1 pt) Is A invertible? Justify your answer. (You do not need to do any calculations to answer this question)   1 2  (b) (3 pts) Prove that x = 1  is an eigenvector of A and find its corresponding 1 eigenvalue λ2 . (c) (2 pts) Use parts (a) and (b) to find the three eigenvalues of A. (d) (6 pts) Is A diagonalizable? (Justify your answer) if so, find a nonsingular matrix S and a diagonal matrix Λ such that A = S ΛS −1 .   0 2 5. (7 points) Let A = . 2 0 (a) (4 pts) Orthogonally diagonalize A, i.e., find an orthogonal matrix Q and a diagonal matrix Λ such that A = QΛQT . (b) (3 pts) Use part (a) to find Ak , where k is a positive integer. (Your answer should be a single matrix). 6. (7 points) Let B be the 2 × 2 matrix   λ1 = 2, λ2 = 0, and with   with eigenvalues 1 1 corresponding eigenvectors x1 = . and x2 = −1 0 (a) (1 pt) Is B diagonalizable? i.e. is there a nonsingular matrix S and a diagonal matrix Λ such that A = SΛS −1 . Justify your answer. (b) (4 pts) Find B . (c) (2 pt) Is B orthogonally diagonalizable? i.e. is there an orthogonal matrix Q and a diagonal matrix Λ such that A = QΛQT . Justify your answer. 7. (a) Find the equation of the straight line y = a0 + a1 t which best fits the following points: t 1 2 3 4 y 1 2 2 3 (b)  Find 1  1   1 1

the projection matrix onto C(A), the column space of A, where A = 1 2  . 3  4       1 2 1       1 2  2   3  1  8. (5 points) Let x1 =   0 , x =  1  , x =  2  . 0 1 0

3 (a) (1 pt) Are x1 , x2 , x3 linearly independent? Justify your answer. (b) (4 pts) Use Gram-Schmidt orthogonalization process to find an orthogonal basis for Span({x1 , x2 , x3 }).   1  1  1 2 3  (c) (2 pts) Find the coordinates of the vector v =   1  w.r.t the basis {u , u , u } 1 obtained in part (b). (d) (1 pts) Find an orthonormal basis for Span({x1 , x2 , x3 }).       1 2 3  2   3   1   2   3   9. (5 points) Let x1 =   0 , x =  1  , x =  1  . 0 1 1 (a) (1 pt) Are x1 , x2 , x3 linearly independent? Justify your answer.       3 1 2 1 1 0 3 2 1 }. Find a basis ,A = ,A = (b) Let S = span of {A = 3 1 2 1 1 0 of S . 10. (5 points) Let the characteristic polynomial of a matrix A be χA (λ) = (−1−λ)2 (3 −λ). (a) (1 pt) What is the order (size or dimension) of A? (b) (2 pts) What is the trace of A? (c) (2 pts) Is A nonsingular (invertible)? Justify your answer. 11. (6 points) 

 x1 (a) (3 pts) Let S be the set of vectors x =  x2  in R3 such that x2 = 5x1 and x3 x3 = 0. Is S a subspace of R3 ? Justify your answer. If S indeed is a subspace of R3 , then find a basis of S and the dimension of S .   x1 (b) (3 pts) Let S2 be the set of vectors x = in R2 such that x1 ≥ 0 and x2 x2 ≤ 0. Is S2 a subspace of R2 ? Justify your answer. If S2 indeed is a subspace of R2 , then find a basis of S2 and the dimension of S2 .   1 α , where α is a scalar. 12. (6 points) Let A = 0 2 (a) (1 pt) For what values of α is A nonsingular (invertible)? Justify your answer. (b) (2 pt) For what values of α is A diagonalizable? Justify your answer. (c) (3 pt) For what values of α is A orthogonally diagonalizable? i.e., A = QΛQT where Q is an orthogonal matrix and Λ is a diagonal matrix. Justify your answer. 13. (6 pts) Let λ1 , . . . , λn be the eigenvalues of matrix A. Assume that A is diagonalizable.

4 (a) Prove that (λ1 )2 , . . . , (λn )2 are the eigenvalues of matrix A2 . (b) Prove that detA = λ1 · · · λn . 14. (10 points) Every part of this problem is independent. (a) (3 points) Let q 1 , q 2 , q 3 be an orthonormal basis of R3 and let x = α1 q 1 + α2 q 2 + α3 q 3 , where α1 , α2 , α3 are scalars. Find α1 in terms of x, q 1 , q 2 , q 3 . (b) (3 pts) Let A and B two n × n matrices. Is it true or false that det(AB) = det(BA)? Justify your answer. (c) (4 pts) Let A and B be two 3 × 3 matrices such that det(A) = 2 and det(B) = 3. Let C = AT (2B)−1 . Find det(C ). (d) Prove that N (A) = N (AT A), where N (A) is the null space of A. 15. Consider the transformation T : R2 → R2 defined by     x1 − x2 x1 . )= T( −x1 + x2 x2 (a) Determine whether or not T is linear. Justify your answer. (b) Find the matrix representation of T w.r.t standard basis in R2 .     1 1 (c) Find the matrix representation of T w.r.t basis B = { } of R2 . , −1 1...