MATH 262: Exam I Solution

25 15 2 3 25 15 4 NAME: 5 20 M262 Exam I. 100 Please show all your work, unless explicitly instructed not to do so. 135 1. (40 points) Let A= 151 001 (a) (10 points) Find RREF(A). E1-E2 0.24 3 35 E,-3E2 THE 001 001 001 E1-11E, 10 = E2+2F3 RREF(A) 0 0 0 (b) (5 points) What is the rank of A? rank of A=3 because 3 leading Is in RREF(A) 0 (c) (10 points) Find all solutions to AT : 0 Ax = 0 13510 100 15110 refer to 1(a) RREF(A)= 010 001 0011 10018 01010 unique solution G=

Still letting A = III 135 001 2 (d) (5 points) Evaluate A 0 1 3x3 3x1 A()]= 001 2 3 7 3 3 (e) (10 points) Is 2 a linear combination of the vectors 1 5 and ? Why or why not? 1 0 0 A ! ! 00 15 irreter to 1(a) RREF(A) = unique solution rank: 3 Ax - b where 6 can be anything 3 = b3 Since the system is consistent and has unique solution, [ is linear combination of vectors HJ@ 3 and

2. (40 points) Consider the function: T:R2-R3 H 2x+y (a) (5 points) Find T ([1]] = 1+4(2) THE 3(1) = 2(1)+2 (b) (10 points) The function T is a linear transformation Find the associated matrix A such that T() = Av. T(V)=Av 14 2 I 4 0 I (c) (10 points) Is this linear transformation onto? That is, is it always true that Ax = b has a solution? A= 3 I 0 4 0 12 7 4 E2 E-4E2 00 10 Arisbanditi RREF(A) 010 001 No solution This linear tranformation is not onto. .

Still letting T:R2-R3 I+4y 2r+y (d) (5 points) Let it and w be R2 = 1 3 2 and mm-[d] 1 two vectors in such that Evaluate T(2+3u). T(2v+3w) = T(27)+T(32) =2T(3)+37(W) 3 2 6+0 2+12 = 7 14 4 4 3 14 (e) (10 points) Let I be a vector in R2 such that T(F) = 6 How many other vectors It are there in R2 with 7 this property (i.e. that by T(F) = = 14 7 42-6 I 4119 3E - E2 41 14 3 0 1 6 12 36 ziz 2E-F3 0 7 2 1 2 21 E2 4114 41 14 3 1 E3 0 0 3 7 E,-4E2 I 002 Urique [H][[]]]] Solution One vector such that [ ]]=[3]

3. (20 points, 4 points each) Circle True or False. You do not need to show your work. (a) True False Let T : R2-R2 be a linear transformation that flips vectors across the line y=x, and let M be the 2 x 2 matrix such that T(F) = Mr. then M2 = I, the identity matrix. (b) True False N=[1]]] invertible. 3-(-1)#0 (ii) (c) True False Let P be a projection onto the x-axis. P() = P(P()). (d) True False A and B are 2 x 2 matrices. It is always true that AB = BA. (e) True False Let P and Q be two perpendicular lines in R2 Then projp(F) + projq(F) = I