Department

MathematicsCourse Code

MATH 33AHProfessor

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FinalThis

**preview**shows pages 1-2. to view the full**7 pages of the document.**MATH 33A LECTURE 3

PRACTICE MIDTERM I

Please note: Show your work. Correct answers not accompanied by suﬃcent explana-

tions will receive little or no credit (except on multiple-choice problems). Please call one

of the proctors if you have any questions about a problem. No calculators, computers,

PDAs, cell phones, or other devices will be permitted. If you have a question about the

grading or believe that a problem has been graded incorrectly, you must bring it to the

attention of your professor within 2 weeks of the exam.

#1 #2 #3 #4 #5 Total

TA name (circle):

Your section meets (circle): Ioannis Lagkas-Nikolos

Tuesday Thursday Jaehoon Lee

Austin Christian

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2016 UC Regents/Dimitri Shlyakhtenko. No part of this exam may be reproduced by any

means, including electronically.

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MATH 33A LECTURE 3 PRACTICE MIDTERM I 2

Problem 1. (True/False, 1 pt each) Mark your answers by ﬁlling in the appropriate box

next to each question (no explanations are necessary on this problem).

(a T F ) There exists a 2 ×2 matrix Aso that A 6=0, A · A 6=0 but A · A · A =0. NO.

Since Ais 2 ×2, ker Ais a subspace of R2and can have dimension 0, 1 or 2. If

dim ker A=2, A=0; but we are given that A 6=0. So ker Acan be dimension 0

or dim 1. If ker Ahad dimension 0, the nullity of Awould be zero, so the rank of A

would be 2 by the rank-nullity theorem. So Awould invertible, but then A3would

also be invertible and so could not be zero. So ker Amust be one-dimensional.

By rank-nullity also im Ais then also one-dimensional.

Let’s say that {v} is a basis for im A. Since Av is in the image of A,Av must be a

multiple of v:Av =αv.

If α 6=0, we would get that A3v=A · A · Av =A · Aαv =Aα2v=α3v 6=0,

contradicting A3=0. So αmust be zero.

If α=0, then Av =0. Let wbe any vector. Then Aw is in the image of A, thus

a multiple of v. Then A·Aw must be a mutliple of Av, which is zero. So A2=0,

again a contradiction. Thus such an Acannot exist.

(b T F ) There are two 2 ×2 matrices A,Bso that AB 6=BA.YES. Take for example A=

0 0

0 1

,B=

0 1

1 0

. Then Ae1=0, Ae2=e2,Be1=e2so ABe1=Ae2=e2

but BAe1=0 because Ae1=0. So ABe16=BAe1so AB 6=BA.

(c T F ) If Ais an n × n matrix with rank A=3 then the system of equations Ax =0

always has exactly one solution. NO. The solution would be unique exactly when

rank A=n; so if n > 3 there would be more than one solution.

(d T F ) If Ais an n×m matrix and im A=Rn, then n ≤ m.YES. The image is spanned

by mvectors Ae1,...,Aem, so by assumption Rnis spanned by these mvectors.

Since dim Rn=nit must be that m ≥ n.

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