Question

5. Which of the following sets of 2 × 2 matrices are vector spaces? (Here ⊕ and ⊙ are the usual addition and scalar multiplication of matrices.) (i) {all matrices [a b; c d] such that a + b = 3c – d.} (ii) {all 2 × 2 matrices A such that Ax = 0 has a nontrivial solution.} (iii) {all 2 × 2 matrices A such that Ax = [1; 2] is consistent.} A. (i) and (ii) B. (ii) only C. (ii) and (iii) D. (i) only E. All of them are vector spaces.

          5. Which of the following sets of 2 &#215; 2 matrices are vector spaces? (Here &#8853; and &#8857; are the usual addition and scalar multiplication of matrices.)
(i) {all matrices [a b; c d] such that a + b = 3c &#8211; d.}
(ii) {all 2 &#215; 2 matrices A such that Ax = 0 has a nontrivial solution.}
(iii) {all 2 &#215; 2 matrices A such that Ax = [1; 2] is consistent.}
A. (i) and (ii)
B. (ii) only
C. (ii) and (iii)
D. (i) only
E. All of them are vector spaces.

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5. Which of the following sets of 2 #215; 2 matrices are vector spaces? (Here #8853; and #8857; are the usual addition and scalar multiplication of matrices.)
(i) all matrices [a b; c d] such that a + b = 3c #8211; d.
(ii) all 2 #215; 2 matrices A such that Ax = 0 has a nontrivial solution.
(iii) all 2 #215; 2 matrices A such that Ax = [1; 2] is consistent.
A. (i) and (ii)
B. (ii) only
C. (ii) and (iii)
D. (i) only
E. All of them are vector spaces.

Added by Hugo G.

Calculus: Early Transcendentals

James Stewart 8th Edition

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Solved by Expert Vincenzo Zaccaro

02/20/2023

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In this case, the matrices are all square and have the same number of columns and rows. Show more…

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5. Which of the following sets of 2 x 2 matrices are vector spaces? (Here and are the usual addition and scalar multiplication of matrices: (i) {all matrices [a b; c d] such that a + b = 3c - d.} (ii) {all 2 x 2 matrices A such that Ax = 0 has a nontrivial solution.} (iii) {all 2 x 2 matrices A such that Ax = [1; 2] is consistent.} A. (i) and (ii) B. (ii) only C. (ii) and (iii) D. (i) only E. All of them are vector spaces.

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