Math 215 Section 2 Final Exam December 20, 1995

Your Name:

If you want to have your exam score and final grade posted, give a four letter code below. Your Code:

The number in parentheses before each question gives its point value. The total is 200 points. Unless otherwise noted, to get full credit, you must show your work and explain your reasoning.

1. (20) True or False? (No reason need be given.)

a) If two augmented matrices are row equivalent, then the associated systems of linear equations have the same solution set.

b) If a system of linear equations has two different solutions, it must have infinitely many solutions.

c) An underconstrained system (one with fewer equations than variables) is always consistent.

d) A system of linear equations has infinitely many solutions if and only if at least one column in the coefficient matrix does not contain a pivot position.

2. (20) Let v_{1} = [ \begin{matrix}1 \\ 0 \\ −1 \end{matrix} ], v_{2} = [ \begin{matrix}−2 \\ 1 \\ −2 \end{matrix} ], v_{3} = [ \begin{matrix}−7 \\ 3 \\ −5 \end{matrix} ], and w = [ \begin{matrix}2 \\ −2 \\ 6 \end{matrix} ].

a) Is w in the span of v_{1}, v_{2}, and v_{3}?

b) Are v_{1}, v_{2}, and v_{3} linearly independent?

c) Give a basis for the space spanned by v_{1}, v_{2}, v_{3}, and w.

3. (20) Let T : P_{2} → P_{2} be the linear transformation that maps a polynomial f(x) to the polynomial (x + 1)f′(x) − f(x). (Here f′ means the derivative of f with respect to x. Make sure you understand the definition of T. As a test, check that T (x) = 1.)

a) Compute the matrix of T with respect to the basis {1, x, x^{2}} of P_{2}.

b) Find a basis for the kernel of T.

c) Find a basis for the image of T.

4. (20) Suppose C is an n × n matrix.

a) If the equation Cx = y has more than one solution for some vector y ∈ R^{n}, do the columns of C span R^{n}? Give a reason for your answer.

b) If the equation Cx = y is inconsistent for some y ∈ R^{n}, is it possible that Cx = 0 has a unique solution? Again give a reason.

5. (20) Use determinants to find all values of h such that the matrix

( \begin{matrix}−1 & 0 & h \\ 0 & 2 & 3 \\ h & 3 & 2 \end{matrix} )

is not invertible.

6. (20) Let v_{1} = [ \begin{matrix}7 \\ 5 \end{matrix} ], v_{2} = [ \begin{matrix}−3 \\ −1 \end{matrix} ], w_{1} = [ \begin{matrix}1 \\ −5 \end{matrix} ], and w_{2} = [ \begin{matrix}−2 \\ 2 \end{matrix} ].

a) Compute the coordinates of [ \begin{matrix}1 \\ 1 \end{matrix} ] with respect to the basis B = \{v_{1}, v_{2}\}.

b) Compute the change-of-coordinates matrix from B = {v_{1}, v_{2}} to C = {w_{1}, w_{2}}.

c) Use a and b to compute the coordinates of [ \begin{matrix}1 \\ 1 \end{matrix} ] with respect to the basis C. Be sure to check directly that your answer is correct.

7. (20) Let A = ( \begin{matrix}9 & −3 \\ −3 & 1 \end{matrix} ).

a) Find an orthonormal basis of R^{2} consisting of eigenvectors of the matrix A.

b) Use your answer to a to find a factorization A = P DP^{−1} where D is diagonal and P is orthogonal.

8. (20) Let Q be the quadratic form on R^{2} whose matrix is ( \begin{matrix}1 & 1 \\ 1 & −1 \end{matrix} ).

a) Compute Q([ \begin{matrix}1 \\ 0 \end{matrix} ]), Q([ \begin{matrix}0 \\ 1 \end{matrix} ]), and Q([ \begin{matrix}1 \\ 1 \end{matrix} ]).

b) Is Q positive definite, negative definite, or indefinite?

9. (20)

a) If the null space of a 7 × 6 matrix A is 5-dimensional, what is the dimension of the column space of A?

b) Is it possible for a nonhomogeneous system of seven equations in six unknowns to have a unique solution?

c) If A is a 4 × 3 matrix, what is the largest possible dimension of the row space of A? What is the largest possible dimension of the column space of A?

10. (20) Find the coefficients β_{0}, β_{1} of the least squares line y = β_{0} + β_{1}x that best fits the following data

points:

(−1, 0), (0, 1), (1, 2), (2, 4)