110.201 LINEAR ALGEBRA 17 December 1997

Final Examination

Three hours. Closed book. No notes. 200 points, 20 per question.

Partial credit may be available, but only if you show your working.

Begin each of the ten questions on a new page and number it clearly in the margin. If you use two books, label them “Book 1 of 2” and “Book 2 of 2”. (If you use

three books, …)

Make sure your T.A.’s name is on each book, as well as your name.

Calculators are allowed but not needed or recommended. Do not give decimal approximations for square roots or trigonometric functions; leave them as is.

There are many opportunities for you to check your work.

1. (a) Find an orthogonal matrix P such that D = P^{−1}AP is a diagonal matrix, where

A = [ \begin{matrix}3 & 5 \\ 5 & 3 \end{matrix} ]

(b) Write down the resulting matrix D (for your choice of P).

2. (a) Find the lengths of the edges of the triangle in R^{4} with vertices P = (2, 0, 1, 2), Q = (4, 1, −1, −1), and R = (1, 5, 1, 3).

(b) Show that P Q is orthogonal to P R.

(c) State Pythagoras’s Theorem for this triangle, and check in writing that it holds.

3. (a) Find a basis of the row space of the matrix

B = [ \begin{matrix}1 & 2 & 3 & 1 & 2 & 3 \\ 3 & 2 & 1 & 3 & 2 & 1 \\ 4 & 5 & 6 & 4 & 5 & 6 \\ 6 & 5 & 4 & 6 & 5 & 4 \end{matrix} ]

(b) Find a basis of the column space of the matrix B.

4. Solve the linear system

\{ \begin{matrix}x & −y & +2z & −2t & = & 1 \\ 2x & −2y & +z & +2t & = & 5 \\ x & −y & +z & & = & 2 \end{matrix}

OVER →

5. (a) Find the point L in the plane 2x + y − 2z = 3 in R^{3} that is closest to the point K = (3, 6, 2). (Hint: Do not use calculus.)

(b) Find the angle between OK and LK (where O denotes the origin).

6. In each part, either write down one example of a set of vectors in R^{4} that satisfies the requested conditions, or give a reason why no such set exists.

(a) A set S_{2} of two nonzero vectors in R^{4} that is not linearly independent.

(b) A set S_{3} of three vectors in R^{4} that is linearly independent.

(c) A set S_{4} of four vectors that spans R^{4} but is not linearly independent.

(d) A set S_{5} of five vectors in R^{4} that is linearly independent.

(e) A set S_{6} of six vectors, all different, that spans R^{4}.

7. (a) Find the eigenvalues of the matrix

C = [ \begin{matrix}4 & c & 2 \\ 0 & 2 & 0 \\ −1 & 3 & 1 \end{matrix} ]

where c is unknown.

(b) For what value(s) of c is the matrix C diagonalizable?

8. Denote by P_{4} the vector space of all polynomials in x of degree 4 (or less), and consider the linear transformation T: P_{4} → P_{4} given by

T (f(x)) = f′(x) + f′′(x)

where f(x) is any polynomial in P_{4}. (Yes, f′ denotes the derivative of f.) Hint: First calculate T (a_{4}x^{4} + a_{3}x^{3} + a_{2}x^{2} + a_{1}x + a_{0}).

(a) Find a basis of the kernel of T.

(b) Find a basis of the range of T.

(c) What is the rank of T?

(d) What is the nullity of T?

9. By any method, find the inverse of the matrix

F = [ \begin{matrix}0 & 2 & 1 \\ 0 & 2 & 2 \\ 1 & 9 & 4 \end{matrix} ]

10. Apply the Gram–Schmidt process to find an orthonormal basis of the subspace W of R^{4} spanned by the vectors (1, −1, 1, 1) and (1, −1, −2, 2). (Hint: Treat the vectors in this order, and the answer will not require any square roots.)

THIS QUESTION SHEET MAY BE RETAINED JMB