MATH70 final Math70-Final-Spring06

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Published on 31 Jan 2019
School
Tufts University
Department
Mathematics
Course
MATH-0070
Professor
Math 46
Final Exam
Spring 2006 Your name
Directions: For each problem, place the letter choice of your answer in the spaces provided on this page.
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1. Given that AB, where
A=
1 4 0 2 1
3 12 1 5 5
2 8 1 3 2
5 20 2 8 8
and B=
140 20
0011 0
000 01
000 00
,
the dimension of Nul(A) is
(a) 1 (b) 2 (c) 3 (d) 4 (e) 5
2. Suppose v1,v2,v3,v4,v5Vand 2v1+ 3v3v4=0.
I. v1,v3,v4are linearly dependent.
II. v1,v2,v3,v4,v5are linearly dependent.
III. v2,v5are linearly independent.
(a) Only I is true
(b) Only II is true
(c) I and II are true
(d) Only III is true
(e) I and III are true
3. Let U= x
yIR2|x+ 2y= 0. Let V= x
yIR2|y= 2x2.
(a) Only Uis a subspace of IR2
(b) Only Vis a subspace of IR2
(c) Both Uand Vare subspaces of IR2
(d) Neither Unor Vis a subspace of IR2
4. Let T: IR3IR5be a linear transformation with T
1
2
3
=
0
0
0
0
0
. Then
(a) Tis 1-1 but not onto
(b) Tis onto but not 1-1
(c) Tis 1-1 and onto
(d) Tis neither 1-1 nor onto
(e) There is not enough information to decide
5. Suppose Ais 10 ×12 and T(x) = Ax. Suppose dim(ker(T)) = 3.
ITis not 1-1
II. Tis not onto
(a) Only I is true
(b) Only II is true
(c) I and II are both true
(d) There’s not enough information to decide
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Document Summary

Directions: for each problem, place the letter choice of your answer in the spaces provided on this page. By signing here, i pledge that i have neither given nor received assistance on this exam. 1: given that a b, where. Ii. v1, v2, v3, v4, v5 are linearly dependent. T is not onto (a) only i is true (b) only ii is true (c) i and ii are both true (d) there"s not enough information to decide. 3: warning: look at the elements of s very carefully. x y z. 7 (b) 9 (c) 16 (d) 63: let a and b be matrices. Suppose t1 : ir3 ir5 is given by t1(x) = ax and t2 : ir2 ir3 is given by. 1: for the eigenvalue = 1 of the matrix a = . , the eigenspace is k-dimensional, where k = (a) 1 (b) 2 (c) 3 (d) 1 is not an eigenvalue of a.