Math 308L WINTER 2014 Final Exam March 20, 2014

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1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 Total 80

• The exam is 110 minutes long. There are 8 problems. Read all of the problems carefully before starting to work on them. • You are allowed to use an 8.5 × 11 sheet of handwritten notes (you may use both sides). You may use a calculator that does not handle matrices and systems of linear equations. No electronic devices (cell-phones, tablets etc) or graphing calculators are allowed. • Show all of your work! You will receive no credit for a problem where you just provide an answer and do not show your work. If you need more room, use the backs of pages. If you do so, please make a note for the grader (for example an arrow and a comment along the lines of “continued on back”). • Write legibly and in an organized manner. If your work is not in a sequential manner, then I will not be able to follow it and hence will not be able to assign any credit. • Box your ﬁnal answer. • If you have any questions, please raise your hand. • Good luck! Math 308E - Summer 2013 Final Exam Page 2 of 11

0.5 0.5 1. Let A = . The matrix A has two eigenvalues λ and λ . The eigenspace E is 0.5 0.5 1 2 λ1 2 2 spanned by ~u = . The other eigenspace E is spanned by ~u = 1 2 λ2 2 −2

(a) (3 points) Find λ1 and λ2.

2 (b) (4 points) Let ~x = . Let k ≥ 1 be an integer. Find a formula for Ak~x . 0 4 0

k (c) (3 points) Find lim A ~x0. k→∞ Math 308E - Summer 2013 Final Exam Page 3 of 11

2. (10 points) Let V = Span(S) be a vector space, where

 2   2   2     2   −2   4  S =   ,   ,   .  0   0   3     0 0 5 

The set S is a linearly independent set of vectors, ( i.e. S is a basis for V ).

Use the Gram Schmidt process to ﬁnd an ORTHONORMAL basis B = {~w1, ~w2, ~w3} for V . Math 308E - Summer 2013 Final Exam Page 4 of 11

2 3 3. Let T : R → R be a linear transformation satisfying  2   −1  2 2 T = 0 T = 0 2   −2   10 −5

1 (a) (2 points) Find T . 0

0 (b) (2 points) Find T . 1

(contd. - see next page) Math 308E - Summer 2013 Final Exam Page 5 of 11

(d) (2 points) Find the dimension of Null space(T ). Math 308E - Summer 2013 Final Exam Page 6 of 11

 1 2 −1 1   −1 0 2 −2  4. Let B =  .  3 6 1 1  2 4 −2 5

(a) (6 points) Find an upper triangular matrix U such that U is row equivalent to B and det(B) = det(U). .

(b) (4 points) Find det(B). Math 308E - Summer 2013 Final Exam Page 7 of 11

 1 −2 −1  5. (a) (5 points) Let A =  0 1 2 . Find a basis for the following vector space 0 0 1

V = { ~x | A~x = AT ~x }. Math 308E - Summer 2013 Final Exam Page 8 of 11

3 4 1 0 −43 132 (b) (5 points) Let M = and D = and B = MDM −1 = . 1 1 0 −10 −11 34 Compute D4 and B4. (Hint : Is it easier to compute D4 ﬁrst ? Can you compute B4 using D4 and M somehow?) Math 308E - Summer 2013 Final Exam Page 9 of 11

3 2 6. (a) (8 points) Construct a 2 × 3 matrix F such that the linear transformation T : R → R given by

3 2 T : R → R , T (~x) = F~x,

satisﬁes the following properties

 2   2  2 3 T 2 = ,T −2 = .   4   5 0 0

 2  (b) (2 points) Using the linear transformation T you constructed in part(a), ﬁnd T  4 . 1 Math 308E - Summer 2013 Final Exam Page 10 of 11

7. Suppose G is a 4×4 DEFECTIVE matrix with characteristic polynomial p(t) = (t−8)3(t−9). Also, suppose that there exists a linearly independent set S = {~u1, ~u2, ~u3} such that ~u1, ~u2 and ~u3 are eigenvectors for A.

a) (4 points) Determine rank(G − 8I4). Justify your answer completely.

b) (3 points) Determine det(G). Provide all the steps.

c) (3 points) Determine the row reduced echelon form of G. Justify your answer completely. Math 308E - Summer 2013 Final Exam Page 11 of 11

8. (I) (3 points) Only one of the following statements is false. Choose the FALSE statement. (i) There exist two matrices A and B such that their characteristic polynomials equal t but the row reduced echelon forms of A and B are not equal to each other. (ii) There exist two matrices A and B such that their characteristic polynomials equal t2 but the row reduced echelon forms of A and B are not equal to each other. (iii) There exist two matrices A and B such that their characteristic polynomials equal t3 but the row reduced echelon forms of A and B are not equal to each other. (iv) There exist two matrices A and B such that their characteristic polynomials equal t4 but the row reduced echelon forms of A and B are not equal to each other.

(II) (4 points) Only one of the following statements is true. Choose the TRUE statement. 2 2 (a) There exists a linear transformation T : 2 → such that T = 1, T = R R 2 −2 2 −1 and T = 0. 4 2 2 (b) There exists a linear transformation T : 2 → such that T = 1, T = R R 2 −2 2 −1 and T = 2. 4 2 2 (c) There exists a linear transformation T : 2 → such that T = 1, T = R R 2 −2 2 −1 and T = −2. 4 2 2 (d) There exists a linear transformation T : 2 → such that T = 1, T = R R 2 −2 2 −1 and T = 4. 4

(III) (3 points) Let A be a 4 × 4 matrix with eigenvalues λ1 = 1 and λ2 = −1. The matrix  2     4  A is not defective. Also, Null space(A + I4) = Span   . Only one of the  3     5  following statements is true. Choose the TRUE statement.

(a) The row reduced echelon form of A + I4 is the identity matrix I4. (b) The algebraic multiplicity of the eigenvalue λ2 equals 2. (c) The geometric multiplicity of the eigenvalue λ1 equals 1. (d) The dimension of Null Space(A − I4) equals 3.