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Suppose a, a, a, and a are vectors in R³, A = (ª₁ | 2₂ | 23 | ª₁), and rref(A) = a. Select all of the true statements (there may be more than one correct answer). A. {a₁, az, as} is a linearly independent set OB. (a₁, 3₂, 33, 34} is a basis for R³ C. {a₁, a₂} is a linearly independent set D. span{a₁, a₂} = R³ E. (a₁, 32, 33, 34} is not a basis for R³ F. a₁ and a₂ are in the null space of A G. span{a₁, ₂, ³, 4} = R³ H. {a₁, az, as, as} is a linearly independent set b. If possible, write as as a linear combination of a and a; otherwise, enter DNE. Enter a1 for ai, etc. 23 -4a1+a2 c. If possible, write a as a linear combination of a, a, and as; otherwise, enter DNE a45a1+3a2+a3 [10 01 1 4 00 0 0

Suppose a, a, a, and a are vectors in R³, A = (ª₁ | 2₂ | 23 | ª₁), and rref(A) = a. Select all of the true statements (there may be more than one correct answer). A. {a₁, az, as} is a linearly independent set OB. (a₁, 3₂, 33, 34} is a basis for R³ C. {a₁, a₂} is a linearly independent set D. span{a₁, a₂} = R³ E. (a₁, 32, 33, 34} is not a basis for R³ F. a₁ and a₂ are in the null space of A G. span{a₁, ₂, ³, 4} = R³ H. {a₁, az, as, as} is a linearly independent set b. If possible, write as as a linear combination of a and a; otherwise, enter DNE. Enter a1 for ai, etc. 23 -4a1+a2 c. If possible, write a as a linear combination of a, a, and as; otherwise, enter DNE a45a1+3a2+a3 [10 01 1 4 00 0 0

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section: Chapter Questions
Problem 1RQ
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Suppose a, a, a, and a are vectors in IR³, A = (a₁ | a₂ | aş | a4), and rref(A) = a. Select all of the true statements (there may be more than one correct answer). A. {a₁, az, as} is a linearly independent set B. {₁, ₂, as, a} is a basis for R³ C. {₁, ₂} is a linearly independent set D. span{a₁, a₂} = R³ E. {a₁, az, as, a} is not a basis for R³ F. a₁ and a₂ are in the null space of A 0 -4 1 0 1 1 4 00 0 0 G. span{a₁, az, aș, a4} = R³ H. {a₁, az, as, a4} is a linearly independent set b. If possible, write as as a linear combination of a₁ and a2; otherwise, enter DNE. Enter a1 for a₁, etc. a3 = -4a1+a2 c. If possible, write a as a linear combination of a, a, and as; otherwise, enter DNE a4 = 5a1+3a2+a3 d. The dimension of the column space of A is 2 and the column space of A is a subspace of R^3 e. Find a basis for the column space of A. Enter your answer as a comma separated list of vectors. Each vector should have the form <a,b,c> or <a,b,c,d> where a,b,... are numbers. Do not use the symbols aj, az, ... in your answers. A basis for the column space of A is {
Transcribed Image Text:Suppose a, a, a, and a are vectors in IR³, A = (a₁ | a₂ | aş | a4), and rref(A) = a. Select all of the true statements (there may be more than one correct answer). A. {a₁, az, as} is a linearly independent set B. {₁, ₂, as, a} is a basis for R³ C. {₁, ₂} is a linearly independent set D. span{a₁, a₂} = R³ E. {a₁, az, as, a} is not a basis for R³ F. a₁ and a₂ are in the null space of A 0 -4 1 0 1 1 4 00 0 0 G. span{a₁, az, aș, a4} = R³ H. {a₁, az, as, a4} is a linearly independent set b. If possible, write as as a linear combination of a₁ and a2; otherwise, enter DNE. Enter a1 for a₁, etc. a3 = -4a1+a2 c. If possible, write a as a linear combination of a, a, and as; otherwise, enter DNE a4 = 5a1+3a2+a3 d. The dimension of the column space of A is 2 and the column space of A is a subspace of R^3 e. Find a basis for the column space of A. Enter your answer as a comma separated list of vectors. Each vector should have the form <a,b,c> or <a,b,c,d> where a,b,... are numbers. Do not use the symbols aj, az, ... in your answers. A basis for the column space of A is {
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