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Unformatted text preview: MATH 33A: MidTerrn I 3 Name:  i4 Q37} Signature:
SID number:
What is the most interesting thing you have learned in this class so far? Circle your section: 1A (Justin Shih, T 9am) 113 (Justin Shih, R 9am)
10 (Joshua Baron, T 9am) 1D (Joshua Baron,R 9am)
1E (Judah Jacobson T 9am) 1F (Judah Jacobson R 9am)
Instructions:
Show all work to receive full credit and feel free to use the back of each paper. No
calculators, references, or formula sheets allowed and please box your answers. You
will receive 1 extra credit point for ﬁlling out all the information above. Problem Possible 8% 1 15 
12
21 2
3
4
5
6 EC
Total Problem 1 (15 ptS) Let A be an n X 71 matrix. List four equivalent statements to:
“The column vectors of A form a basis of R“.” Am; 33? Mar gm” Mei/2 Wo‘fr‘jQ/i :3 aim (333:3! iii/‘3 ﬁdmmaﬂf 31m om {)3 i354 Let A be an n X nmatrix, B be an m X 71 matrix, and C be an n X m matrix, 17
be an n X 1 vector and if be an 1 X n vector, where n 3% m. Which of the following
expressions are well deﬁned? No work is need here for full credit. 1' 03 be) 0‘3“ well Gltfﬂa egg/i 2. 6 a W all at eﬂlm r: at 3. ABC M353» Well c/ire'ffmm ml
4. an Well {Jl Seam 525;! 5. Br; Wei! 0i (gm/l 6. 03M Maj“ wail Criéiﬂﬂﬁf‘f 7. «3 U 'W all [legal a... ProbEem 2 (12 pts) If possible, ﬁnd the inverse of what is the rank of A? "I; 92;. E
#25” L ”a J 0 ‘_ C3 {3 ‘
g 92 <2: Problem 3 (21 pts) Part 1: Please construct a 2 X 2 matrix A which rotates a vector by 90° clockwise. Next construct a matrix C which projects all vectors in R2 onto the line y : 2m. 953531.};in bl; ngti 9 £13" yin/{*7 {7; [ 2:331:57?" _ 4} ﬂm 0g? @‘WWGQQ W5. 60} Part II: Carefully plot and label what happens geornetrieally to 51 and é}; under the
linear transformations M1 = A0 and M2 = CA Le. plot Mﬁé’i) and M2(é;) for
1} = 1,2. What can we conclude about the commutativity of A and C"? Note: No
computation is necessary to answer Part II. Problem 4 (22pts) Let A be A: .36~36—6
002—64' 1. What is the domain and range of the linear transformation Th?) = A5? 2.. Find rref(A). 3. Finé a basis for Im(A). 4. Find a basis for Ker(A). @ Hymn a; [j J J «J J J @ (59353 71;;
(J 331 f {355% fBJWCJ’ 2 35%
1/3
' \j 1,.wa
3 39 “/J I, 4?
3% J: J»; 3’” 1:} 1*? WJ J,” J JJJ JJJ 0 3L 4
J {J JJJ’K LI: “7%szijwa Qééwg‘ [,2 Problem 5 (10pts) Consider the plane 23:1 7 3:32 + 4503 : O with basis l3 vectors 1'11 and ﬁg Where only 111 8
1713 4 .
—1 Find she second basis vector v2 which satisﬁes is known: Problem 6 (20pts) Please state whether the following statements are necessarily true (no work needs to
be shown here). 1. A vertical shear iinear transformation is always invertible. 'ﬂrLifQ 2. If V and W are subspaces of R” such that dirn(V) + dimU/V) : n; then
V U W 3 RH' FTC/1 5‘5an 3. There exists a 2 X 2 matrix A 31$ 0 such that A2 = 0, where “0” is the 2 X 2
matrix of zeros. TVka 4. Let the line L be any line through the origin. Then you can write any vector
:E'ER2 as :75: ro' f +refJ :E’. J?
p tr) A) M/m 5. The domain of the linear transformation T($) = Am, where A is a 20 X 10
. A 20 ,r
matrix, is R . f5??? i331} 6. There is a 3 x 3 invertible matrix where seven of the nine entries are 1. fy‘w '5: 7. If A is an 5 X 5 matrix and you can ﬁnd 3 distinct soiutions to Ass 2 0 then you can always ﬁnd 4. .FYU i
8. A 3 X 2 matrix is never invertible. fmC "". 9. If A is a r x p matrix then dirn(lrn(A)) +dirn(ker(A)) = p. fig/“ti '1: 10. If the set of vectors {v1 . . . Ur} spans R”, then k : n. FE? 11136., ...
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 Spring '09
 DAI,S.

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