Suppose aj, az, az, and a, are vectors in R and they are, in order from left to right, the four columns of the matrix (A), so that A = [aj | a | a | a), and [i 0 1 -31 rref(A) = 0 2 0 0 0 3 a. Select all of the true statements (there may be more than one correct answer). OA. span(ai, a2, az, ā4) = R OB. {ai, a2, ā3, ã4} is a linearly independent set OC. span(ai, az) = R³ OD. {āi, az} is a linearly independent set OE. faj, a2, a3, a} is a basis for 33 OF. {ai, a2, ā3, a4} is not a basis for R³ OG. aj and az are in the kernel of A OH. {ãi, a2, ā3} is a linearly independent set b. If possible, write az as a linear combination of aj and az; otherwise, enter "DNE". You may enter "a1" for aj, etc, or enter coordinate vectors of the form “<1,2,3>" or <1,2,3,4,5>". c. If possible, write a as a linear combination of aj and az; otherwise, enter "DNE". Again, you may enter "a1" for aj, etc, or enter coordinate vectors of the form "<1,2,3>" or "<1,2,3,4,5>". a4 =
Suppose aj, az, az, and a, are vectors in R and they are, in order from left to right, the four columns of the matrix (A), so that A = [aj | a | a | a), and [i 0 1 -31 rref(A) = 0 2 0 0 0 3 a. Select all of the true statements (there may be more than one correct answer). OA. span(ai, a2, az, ā4) = R OB. {ai, a2, ā3, ã4} is a linearly independent set OC. span(ai, az) = R³ OD. {āi, az} is a linearly independent set OE. faj, a2, a3, a} is a basis for 33 OF. {ai, a2, ā3, a4} is not a basis for R³ OG. aj and az are in the kernel of A OH. {ãi, a2, ā3} is a linearly independent set b. If possible, write az as a linear combination of aj and az; otherwise, enter "DNE". You may enter "a1" for aj, etc, or enter coordinate vectors of the form “<1,2,3>" or <1,2,3,4,5>". c. If possible, write a as a linear combination of aj and az; otherwise, enter "DNE". Again, you may enter "a1" for aj, etc, or enter coordinate vectors of the form "<1,2,3>" or "<1,2,3,4,5>". a4 =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Transcribed Image Text:Suppose aj, az, az, and a, are vectors in R3 and they are, in order from left to right, the four columns of the matrix (A), so that A = [aj | az | az | a), and
Г1 0
1
-3
rref(A) = 0 1 2
0 0 0
3
a. Select all of the true statements (there may be more than one correct answer).
OA. span(āi, a2, āz, a4) = R
OB. fāi, a2, a3, a4} is a linearly independent set
OC. span(āi, ā) = R³
OD. {ãi, a2} is a linearly independent set
OE. faj, a2, az, a4} is a basis for R
OF. {ai, a2, a3, a4} is not a basis for R
OG. aj and a, are in the kernel of A
OH. {ãi, a2, ā3} is a linearly independent set
b. If possible, write a as a linear combination of aj and as; otherwise, enter "DNE". You may enter "a1" for ai, etc, or enter coordinate vectors of the form “<1,2,3>" or "
<1,2,3,4,5>".
az =
c. If possible, write a as a linear combination of aj and az; otherwise, enter "DNE". Again, you may enter "a1" for aj, etc, or enter coordinate vectors of the form
"<1,2,3>" or " <1,2,3,4,5>".
a4 =
d. The dimension of the image of A is
and the image of A is
subspace of
(enter "R^n" with a specfic number for "n").
e. Find a basis for the image of A. Enter your answer as a comma separated list of vectors of the form "a1" etc, or of the form "<a,b,c>" or "<a,b,c,d>" where a,b,... are
numbers.
A basis for the image of A is {
}
f. The dimension of the kernel of A is
and the kernel of A is
subspace of
(enter "R^n" with a specfic number for "n").
g. Find a basis for the kernel of A. Enter your answer as a comma separated list of vectors of the form "<a,b,c>" or "<a,b,c,d>" where a,b,... are numbers.
basis for the kernel of A is {
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