Unformatted text preview:

Math 51, Winter 2007 Midterm 2 March 1, 2007MIDTERM 2• Complete the following problems. You may use any result from class you like, but if you cite a theorembe sure to verify the hyp otheses are satisfied.• This is a closed-book, closed-notes exam. No calculators or other electronic aids will be permitted.• In order to receive full credit, please show all of your work and justify your answers. You do not needto simplify your answers unless specifically instructed to do so.• If you need extra room, use the back sides of each page. If you must use extra paper, make sure towrite your name on it and attach it to this exam. Do not unstaple or detach pages from this exam.• Please sign the following:“On my honor, I have neither given nor received any aid on this examina-tion. I have furthermore abided by all other asp ects of the honor code withrespect to this examination.”Name:Signature:1 10 pts2 10 pts3 13 pts4 15 pts5 10 pts6 12 pts7 10 ptsTotal 80 ptsCircle your TA’s nameLanOrenJoshPeterChadLeoRobNikolaJianPage 1 of 10Math 51, Winter 2007 Midterm 2 March 1, 2007(1) (10 point) Find the determinant of each of the following matrices. Show your work or justify youranswer.(a)2 3 1 01 0 2 23 5 6 92 0 3 3(b)1 2 3 5 10 0 32 3 1 1 0 0 23 5 4 6 10 0 50 7 2 3 4 2 14 1 8 9 10 6 73 1 5 1 1 10 32 0 0 3 4 1 8(Hint: What is row 3 in terms of rows 1 and 2?)Page 2 of 10Math 51, Winter 2007 Midterm 2 March 1, 2007(2) (10 points) Suppose that V ⊆ R4is a subspace with basisB =2201,6340.(a) Use the Gram-Schmidt process and B to produce an orthonormal basis for V .(b) Compute the orthogonal projection of−→x =90259onto V .Page 3 of 10Math 51, Winter 2007 Midterm 2 March 1, 2007(3) (13 points) One can show that B = {−→v1,−→v2,−→v3,−→v4} is a basis for R4, where−→v1=1234,−→v2=5678,−→v3=1100,−→v4=1030.(You don’t have to prove this!)(a) Give the matrix C which changes basis from B to the standard basis. That is, find C so thatC[−→v ]B= [−→v ]S,where [−→v ]Bis−→v in B-coordinates and [−→v ]Sis−→v in standard coordinates.(b) Suppose that T : R4→ R4is the linear transformation which satisfies• T (−→v1) =−→v1+ 2−→v2,• T (−→v2) =−→v1+−→v3,• T (−→v3) =−→v1+−→v2−−→v4, and• T (−→v4) =−→0 .Give the matrix for T in coordinates relative to the basis B.Page 4 of 10Math 51, Winter 2007 Midterm 2 March 1, 2007(c) Give the matrix for T in coordinates relative to the standard basis. (You may express your answeras a product of matrices and their inverses without expanding out the products or computing theinverses).(4) (15 points) LetA =1 3 10 5 10 −9 −1.(a) Find all eigenvalues of A.Page 5 of 10Math 51, Winter 2007 Midterm 2 March 1, 2007(b) For each eigenvalue, give a basis for the corresponding eigenspace.(c) Is A diagonalizable? Be sure to explain your answer fully.Page 6 of 10Math 51, Winter 2007 Midterm 2 March 1, 2007(5) (10 points) Suppose that B = {−→v1, · · · ,−→vn} is an orthonormal basis for Rn, and let Q be the (square)matrixQ =| |−→v1· · ·−→vn| |.(a) What is QTQ? Justify your answer.(b) Prove that kQ−→x k = k−→x k.Page 7 of 10Math 51, Winter 2007 Midterm 2 March 1, 2007(6) (12 points) Let V be a k-dimensional subspace of Rnfor some 0 < k < n, and let T be the lineartransformation which projects vectors onto V .(a) Prove that there exists a non-zero vector in V⊥.(b) Prove that 0 is an eigenvalue of T .(c) Prove that 1 is an eigenvalue of T .Page 8 of 10Math 51, Winter 2007 Midterm 2 March 1, 2007(7) (10 points)(a) Letf(x, y) =(|y|, if |x| ≤ |y||x|, if |y| < |x|.Carefully draw (and lab el) the level curves f(x, y) = 0, f(x, y) = 1 and f(x, y) = 4.Page 9 of 10Math 51, Winter 2007 Midterm 2 March 1, 2007(b) Determine the value of the constant c so thatg(x, y) =x3+ xy2+ 2x2+ 2y2x2+ y2if (x, y) 6= (0, 0)c if (x, y) = (0, 0)is continuous. Be sure to justify your answer.Page 10 of


View Full Document

Stanford MATH 51 - Study Notes

Download Study Notes
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Study Notes and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Study Notes 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?