Suppose a1, a2, a3, a4, and ag are vectors in R³, A = (a1 | a2 | a3 | a4 | a5), and [1 0 -5 5 rref(A) 0 1 4 0 0 1 1 -3 a. Select all of the true statements (there may be more than one correct answer). OA. span{a1, az} = R³ MB. {a1, a2} is a linearly independent set OC. {a1, a2, a3, a4} is a basis for R3 OD. span{a1, a2, a3} = R3 VE. span{a1, a2, a3, a4} = R VF. {a1, a2, a3} is a linearly independent set OG. {a1, a2, a3, a4} is a linearly independent set OH. {a1, az} is a basis for R3 VI. {a1, a2, a3} is a basis for R3 b. If possible, write a4 as a linear combination of a1, a2, and a3; otherwise, enter DNE. a4 = -5a1 + 4a2 + a3 c. The dimension of the column space of A is 3 and the column space of A is a subspace of R^3 d. Find a basis for the column space of of A. If necessary, enter a1 for a1, etc., or enter coordinate vectors of the form <1,2,3> or <1,2,3,4,5>. Enter your answer as a comma separated list of vectors. A basis for the column space of A is { a1,a2,a3 e. The dimension of the null space of A is 2 and the null space of A is a subspace of R^5 f. If x1 = (5, –4, –1,1, 0), then Ax1 = [0,0,0] Is x1 in the null space of A? yes g. If x2 = (-5, -2, 3,0, 1), then Ax2 = [0,0,0] Is x2 in the null space of A? yes h. If x3 = 3x2 – 4x1 = [0,0,0] then Ax3 = Is xg in the null space of A? yes i. Find a basis for the null space of of A. If necessary, enter a1 for ai, etc., or enter coordinate vectors of the form <1,2,3> or <1,2,3,4>. Enter your answer as a comma separated list of vectors. A basis for the null space of A is { [5,-4,-1,1,0],[-5,-2,3,0,1]
Suppose a1, a2, a3, a4, and ag are vectors in R³, A = (a1 | a2 | a3 | a4 | a5), and [1 0 -5 5 rref(A) 0 1 4 0 0 1 1 -3 a. Select all of the true statements (there may be more than one correct answer). OA. span{a1, az} = R³ MB. {a1, a2} is a linearly independent set OC. {a1, a2, a3, a4} is a basis for R3 OD. span{a1, a2, a3} = R3 VE. span{a1, a2, a3, a4} = R VF. {a1, a2, a3} is a linearly independent set OG. {a1, a2, a3, a4} is a linearly independent set OH. {a1, az} is a basis for R3 VI. {a1, a2, a3} is a basis for R3 b. If possible, write a4 as a linear combination of a1, a2, and a3; otherwise, enter DNE. a4 = -5a1 + 4a2 + a3 c. The dimension of the column space of A is 3 and the column space of A is a subspace of R^3 d. Find a basis for the column space of of A. If necessary, enter a1 for a1, etc., or enter coordinate vectors of the form <1,2,3> or <1,2,3,4,5>. Enter your answer as a comma separated list of vectors. A basis for the column space of A is { a1,a2,a3 e. The dimension of the null space of A is 2 and the null space of A is a subspace of R^5 f. If x1 = (5, –4, –1,1, 0), then Ax1 = [0,0,0] Is x1 in the null space of A? yes g. If x2 = (-5, -2, 3,0, 1), then Ax2 = [0,0,0] Is x2 in the null space of A? yes h. If x3 = 3x2 – 4x1 = [0,0,0] then Ax3 = Is xg in the null space of A? yes i. Find a basis for the null space of of A. If necessary, enter a1 for ai, etc., or enter coordinate vectors of the form <1,2,3> or <1,2,3,4>. Enter your answer as a comma separated list of vectors. A basis for the null space of A is { [5,-4,-1,1,0],[-5,-2,3,0,1]
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.3: Spanning Sets And Linear Independence
Problem 21EQ
Related questions
Question
Please only answer the red boxes!
![Suppose a1, a2, a3, a4, and az are vectors in R³, A = (a1 | a2 | a3 | a4 | a5), and
[1 0 0 -5
1 0
5
rref(A)
4
[0 0 1
1
-3
a. Select all of the true statements (there may be more than one correct answer).
OA. span{a1, a2} = R³
|B. {a1, a2} is a linearly independent set
)C. {a1, a2, a3, a4} is a basis for R3
D. span{a1, a2, a3} = R³
E. span{a1, a2, a3, a4} = R³
F. {a1, a2, a3} is a linearly independent set
|G. {a1, a2, a3, a4} is a linearly independent set
|H. {a1, a2} is a basis for R3
V1. {a1, a2, a3} is a basis for R³
b. If possible, write a4 as a linear combination of a1, a2, and a3; otherwise, enter DNE.
a4 =
-5а1 + 4а2 + а3
c. The dimension of the column space of A is 3
and the column space of A is a subspace of R^3
d. Find a basis for the column space of of A. If necessary, enter a1 for a1, etc., or enter coordinate vectors of the form <1,2,3> or <1,2,3,4,5>. Enter your answer as a
comma separated list of vectors.
A basis for the column space of A is { a1,a2,a3
e. The dimension of the null space of A is 2
and the null space of A is a subspace of
R^5
f. If x1
(5, –4, –1, 1,0), then Ax1
[0,0,0]
Is x1 in the null space of A?
yes
g. If x2 = (-5, -2, 3,0, 1), then Ax2 =
[0,0,0]
Is x2 in the null space of A?
yes
h. If x3
Зx — 4х1
[0,0,0]
then Ax3
Is x3 in the null space of A?
yes
i. Find a basis for the null space of of A. If necessary, enter a1 for a1, etc., or enter coordinate vectors of the form <1,2,3> or <1,2,3,4>. Enter your answer as a comma
separated list of vectors.
A basis for the null space of A is { [5,-4,-1,1,0],[-5,-2,3,0,1]
}](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F05295cc1-71f0-4e3b-b6f8-2c1731c6d617%2F8c061155-246a-4072-b9f0-ab74dcf893e1%2Fhz83oic_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose a1, a2, a3, a4, and az are vectors in R³, A = (a1 | a2 | a3 | a4 | a5), and
[1 0 0 -5
1 0
5
rref(A)
4
[0 0 1
1
-3
a. Select all of the true statements (there may be more than one correct answer).
OA. span{a1, a2} = R³
|B. {a1, a2} is a linearly independent set
)C. {a1, a2, a3, a4} is a basis for R3
D. span{a1, a2, a3} = R³
E. span{a1, a2, a3, a4} = R³
F. {a1, a2, a3} is a linearly independent set
|G. {a1, a2, a3, a4} is a linearly independent set
|H. {a1, a2} is a basis for R3
V1. {a1, a2, a3} is a basis for R³
b. If possible, write a4 as a linear combination of a1, a2, and a3; otherwise, enter DNE.
a4 =
-5а1 + 4а2 + а3
c. The dimension of the column space of A is 3
and the column space of A is a subspace of R^3
d. Find a basis for the column space of of A. If necessary, enter a1 for a1, etc., or enter coordinate vectors of the form <1,2,3> or <1,2,3,4,5>. Enter your answer as a
comma separated list of vectors.
A basis for the column space of A is { a1,a2,a3
e. The dimension of the null space of A is 2
and the null space of A is a subspace of
R^5
f. If x1
(5, –4, –1, 1,0), then Ax1
[0,0,0]
Is x1 in the null space of A?
yes
g. If x2 = (-5, -2, 3,0, 1), then Ax2 =
[0,0,0]
Is x2 in the null space of A?
yes
h. If x3
Зx — 4х1
[0,0,0]
then Ax3
Is x3 in the null space of A?
yes
i. Find a basis for the null space of of A. If necessary, enter a1 for a1, etc., or enter coordinate vectors of the form <1,2,3> or <1,2,3,4>. Enter your answer as a comma
separated list of vectors.
A basis for the null space of A is { [5,-4,-1,1,0],[-5,-2,3,0,1]
}
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