Name:

1  ^{2}  ^{3}  ^{4}  ^{5}  ^{6}  ^{7}  ^{Σ}  Instructor:

NORTHEASTERN UNIVERSITY Department of Mathematics

MTH 1230  FINAL EXAM  Fall 1997

Instructions: Put your name and the instructor’s name in the blanks above. Put your final answers to each question in the designated spaces—you may lose credit if you don’t. Calculators are permitted. A single cribsheet of formulas is allowed. SHOW YOUR WORK. If there is not enough room to show your work, use the back of the preceding page. Good luck!

1. 8 points Are the following vectors independent or dependent? If they are independent, say why. If they are dependent, exhibit a linear dependence relation among them.

v_{1} = [\begin{matrix}3 \\ 0 \\ −1 \end{matrix}]  v_{2} = [\begin{matrix}0 \\ 3 \\ 5 \end{matrix}]  v_{3} = [\begin{matrix}1 \\ −4 \\ −7 \end{matrix}]

MTH 1230  Final Exam, Fall 1997  Page 2

2. 20 points Let A = [\begin{matrix}2 & 0 & 3 \\ 4 & 5 & 7 \\ −6 & 10 & −7 \end{matrix}], b = [\begin{matrix}−2 \\ −3 \\ 8 \end{matrix}].

(a) Solve the system of linear equations Ax = b, indicating clearly the row operations, pivots, leading variables, and free variables.

(b) Find the LU-decomposition of the matrix A.

(c) Find a basis for each of the row space, column space, and null-space of A.

(d) Find the rank of A.

(e) Find a non-zero vector that is orthogonal to both the vectors [2 0 3] and [−6 10 −7].

MTH 1230  Final Exam, Fall 1997  Page 3

3. 12 points In each of the following, a vector space V and a subset S are given. Circle one answer:

(a) V = ℝ^{4}, S = {(−t, 4t, 3t, 0) | t is a real number}

S is closed under addition:  YES NO
S is closed under scalar multiplication:  YES NO
S is a vector subspace of V:  YES NO

(b) V = ℝ^{4}, S = {(−t, 4t, 3t, 1) | t is a positive real number}

S is closed under addition:  YES NO
S is closed under scalar multiplication:  YES NO
S is a vector subspace of V:  YES NO

(c) V = ℝ^{4}, S = {x ∈ V | x_{1} + 2x_{2} − x_{3} = 0, 2x_{1} + x_{3} − 3x_{4} = 0}

S is closed under addition:  YES NO
S is closed under scalar multiplication:  YES NO
S is a vector subspace of V:  YES NO

(d) V = ℝ^{4}, S = {x ∈ V | x_{1} + 2x_{2} − x_{3} ≥ 0}

S is closed under addition:  YES NO
S is closed under scalar multiplication:  YES NO
S is a vector subspace of V:  YES NO

MTH 1230  Final Exam, Fall 1997  Page 4
4. 12 pts

(a) Find the 3 × 3 matrix A associated with the linear mapping that rotates the yz-plane by −60^{∘} and reflects the x-axis about the yz-plane.

(b) Is A orthogonal?

(c) What is det(A)?

(d) What is the image of the point (−3, 2, 1) under the above mapping?

MTH 1230  Final Exam, Fall 1997  Page 5

5. 12 pts Let A = [\begin{matrix}1 & 2 \\ 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{matrix}] and b = [\begin{matrix}−2 \\ 3 \\ 0 \\ 4 \end{matrix}].

(a) Find the least squares solution [ \overline{x} ; \overline{y} ] of the inconsistent system A · [ \begin{matrix}x \\ y \end{matrix} ] = b.

(b) Use your answer to part (a) to find the projection of b onto CS(A), the column space of A.

MTH 1230  Final Exam, Fall 1997  Page 6

6. 16 pts Let a_{1} = [\begin{matrix}3 \\ 4 \end{matrix}] , a_{2} = [\begin{matrix}−1 \\ 1 \end{matrix}] . (a) Find unit vectors in the direction of a_{1} and a_{2}, respectively.

(b) Find the lengths of a_{1} and a_{2}, and compute the dot product a_{1} · a_{2}.

(c) Find the angle between a_{1} and a_{2}. Are a_{1} and a_{2} orthogonal?

(d) Let A = [a_{1} a_{2}]. Use the Gram-Schmidt process to find the QR-factorization of A.

MTH 1230  Final Exam, Fall 1997  Page 7

7. 20 pts Let A = [\begin{matrix}2 & 2 \\ 3 & 1 \end{matrix}] .

(a) Find the characteristic equation for A.

(b) Find the eigenvalues of A.

(c) Find a basis for each eigenspace of A.

(d) Form a matrix S using the two independent eigenvectors from part (c) as column vectors, and calculate S^{−1}.

(e) Calculate S^{−1} · A · S. Explain your answer.