Math 51, Winter 2007 Midterm 1 February 1, 2007MIDTERM 1• 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:The following boxes are strictly for grading purposes. Please do not mark.1 15 pts2 10 pts3 15 pts4 15 pts5 15 pts6 10 pts7 15 ptsTotal 95 ptsPage 1 of 9Math 51, Winter 2007 Midterm 1 February 1, 2007(1) (15 points) Complete each of the following sentences.(a) A collection of vectors−→v1, · · · ,−→vmis defined to be linearly dependent if(b) The dimension of a subspace W is defined to be(c) The book lists 4 properties a matrix must have to be in reduced row echelon form. Two of theseproperties are(i)(ii)Page 2 of 9Math 51, Winter 2007 Midterm 1 February 1, 2007(2) (10 points) Compute the reduced row echelon form of the matrix1 3 4 01 −1 0 12 3 5 3−2 −1 −3 1.Page 3 of 9Math 51, Winter 2007 Midterm 1 February 1, 2007(3) (15 points) A matrix B and its reduced row echelon form are given below:B =1 2 4 11 72 4 5 13 83 6 6 15 10and rref(B) =1 2 0 −1 00 0 1 3 00 0 0 0 1.(a) Give a basis for the null space of B. You do not need to prove that your collection is a basis.(b) Give a basis for the column space of B. You do not need to prove that your collection is a basis.(c) Find a non-zero vector orthogonal to all three of the vectors124117,245138,3661510.Page 4 of 9Math 51, Winter 2007 Midterm 1 February 1, 2007(4) (15 points) Determine whether each statement is true or false and circle your answer. In all thesestatements, A is an arbitrary m × n matrix and R = rref(A). No justification is necessary for youranswers.T or F (a) N(A) = N (R).T or F (b) C(A) = C(R).T or F (c) The dimensions of C(A) and C(R ) are the same.T or F (d) If the equation A−→x =−→0 has infinitely many solutions, then for any−→b ∈ Rmthe system A−→x =−→bhas infinitely many solutions.T or F (e) If n > m, it is possible for dim(C(A)) = m.T or F (f) If n > m, it is possible for dim(C(A)) = n.T or F (g) If n > m, it is possible for dim(N (A)) = 0.Page 5 of 9Math 51, Winter 2007 Midterm 1 February 1, 2007(5) (15 points) Let P1and P2be two planes passing through the point (1, 1, 1) such that−→n1is normal toP1and−→n2is normal to P2, where−→n1=315and−→n2=102.(a) Find the parametric equation of the line ` which is the intersection of P1and P2.Page 6 of 9Math 51, Winter 2007 Midterm 1 February 1, 2007(b) Prove that P1is not a subspace. (Hint: To answer this question you should find a property ofsubspaces that P1does not have. Be sure you give full justification for any claims you make.)(6) (10 points) Suppose that−→x and−→y are two vectors in Rnwith equal magnitude. Prove that the vectors−→x +−→y and−→x −−→y are orthogonal.Page 7 of 9Math 51, Winter 2007 Midterm 1 February 1, 2007(7) (15 points) Consider a linear transformation T : R4→ R3which satisfiesT1100=111, T0110=012,T0011=210, and T0001=−1−1−1.(a) Compute T1000.Page 8 of 9Math 51, Winter 2007 Midterm 1 February 1, 2007(b) Suppose you are told that a linear transformation S satisfiesS100=0000, S010=1010and S001=51015−5.What is the matrix A that satifies A−→x = S(−→x ) for all−→x ∈ R3?Page 9 of
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