In each of the cases below, is the set B a basis of R"? Explain why or why not. (a) {[2]} (b) n = 1 and B n = 2 and...

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$In each of the cases below, is the set B a basis of R? Explain why or why not. (a) {[2]} (b) n = 1 and B n = 2 and B = _ (c)$

math algebra

Answer 1

Answer #1

(a). B is not a basis for R, the set of real numbers as a singleton matrix is not the same as a number.

(b). The set B is linearly dependent as (1,1)^T = (1,0)^T+(0,1)^T. Hence B is not a basis for R².

( c). Since dim(R³ ) = 3, and since B contains only 2 vectors, hence B is not a basis for R³.

(d). Let A be the matrix with the vectors in the set B as columns. Then A =

The RREF of A is

This implies that (0,5,10)^T= -15(1,1,0)^T+5(3,4,2)^T. Thus, the set B is linearly dependent. Therefore, B is not a basis for R³.

(e). The set B is linearly independent and the vectors in B spans R². Therefore, B is a basis for R².

(f). Let A be the matrix with the vectors in the set B as columns. Then A =

The RREF of A is I₄ which implies that the set B is linearly independent and the vectors in B spans R⁴. Therefore, B is a basis for R⁴.

Answer 2

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