2017~2018 Final Exam 线性代数英文试题

Your Name:

2016—2017 Fall Semester

UNIVERSITY OF SCIENCE ＆ TECHNOLOGY BEIJING

Linear Algebra Final Exam

Time: 09:00-11:30 A.M. Full Mark: 100 Your Mark: Notation: Please fill out and sign the front of your exam booklet.

No books or electronic devices allowed. No using any notes or formulas! No cheating!

You may keep this paper. Solutions will be posted on the course website after the exam.

Please do not answer the following problem until we give the signal.

1. (20 points) Let A = [ 221 6 63−3 011040 13

104

] Find bases of the following vector spaces and state

their dimensions.

(a) The column space of A.

(b) The row space of A.

(c) The null space of A.

(d) The orthogonal complement of the column space of A.

2. (15 points) Let A = (4−130

) (a) Compute A k for all integers k ≥0. Write the answer as explicitly as you can, in the form of a 2×2 -matrix with entries depending on k.

(b) Solve the initial value problem x '(t) = A x (t) with x (0) =(10

)

3. (10 points)

(a) Let P n be the vector space of polynomials of degree less than or equal to n. Let T be the linear transformation from P 3 to P 4defined by

T(p )(t )=p (2) + (t – 2)ṗ(t)+ t 3p̈(5t)

(You are not required to show that T is linear.) Find the matrix of T with respect to the B 3 = {1, t, t 2, t 3} of P3 and the B 4 = {1,t,t 2,t 3,t 4} of. P 4

(b) Find the equation y = ax + b of the least-squares line that best fits the data points (1,2), (2,2), and (3,4).

4. (20 points) Let A= (23

101021

) (a) Find the singular values of A.

(b) Find two unit vectors in R 4 that are orthogonal to each other and to the columns of A. (c) Find a singular value decomposition of A.

5. (15 points) Prove the following assertions. All matrices in this problem are real n ×n -matrices.

(a) If matrices A and B are similar, then they have the same rank.

(b) Suppose the matrix A satisfies the following conditions: A is symmetric, A 2 = A, and rank(A) =

1. Then there exists a unit vector u in R n with the property that A = uu T . (Hint: what does the condition A 2 = A tell you about the eigenvalues of A? Also use the result of part (a).)

6. (20 points) True or false? Prove your assertions! All matrices in this problem are real. (a) If A is an m x n-matrix, then A and A T A have the same null space.

(b) The formula (f,g ) =∫(f (t )+ f’(t ))(g (t )+ g’(t ))1

0dt defines an inner product on the space of continuously differentiable functions on the interval [0,1]. (A function f is called continuously differentiable if f’ exists and is continuous.)

(c) If A is a positive definite symmetric n ×n -matrix, then there exists a non-zero vector x in R n with the property that x T A x > ||x ||2.

(d) If A is an m ×n -matrix and Q is an orthogonal n ×n -matrix, then A and AQ have the same singular values.