2017~2018 Final Eam 线性代数英文试题

Your Name：Student Number:20162017 Fall Semester UNIVERSITY OF SCIENCE TECHNOLOGY BEIJING Linear Algebra Final ExamYour Mark:Time: 09:00-11:30 A.M.Full Mark: 100Notation: Please fill out and sign the front of your exam booklet.Let A = (34No 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.23104. (20 points) Let A= 1 021(a) Find the singular values of A.(b) Find two unit vectors in R4 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 realn X 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, A2 = A, and rank(A) = 1. Then there exists a unit vector u in Rn with the property that A = uuT. (Hint: what does the condition A2 = 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 ATA have the same null space.(b) The formula (f, g) = t + f，t g t + gf t dt 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 X “-matrix, then there exists a non-zero vector x in Rn with the property that xTAx |x|2.(d) If A is an mxn -matrix and Q is an orthogonal” X n -matrix, then A and AQ have the same singular values.26 -3 10131. (20 points) Let A = 2 6 0 4 10 Find bases of the following vector spaces and state1 3 1 0 4their 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)(a) Compute Ak for all integers k0. Write the answer as explicitly as you can, in the form of a 2x2 -matrix with entries depending on k.(b) Solve the initial value problem x(t) = Ax(t) with x(0) =(0* (a) (b) (c) (d) * * * * * * * 1)3. (10 points)(a) Let Pn be the vector space of polynomials of degree less than or equal to n. Let T be the linear transformation from P3 to P4defined byT(p)(t)=p 2) + (t - 2)Q(t)+ t30(5t)(You are not required to show that T is linear.) Find the matrix of T with respect to theB3= 1, t, t2, t3 of P3 and the B 4 = I,t,t2,t3,t4 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).