Consider the following subsets of R^3 I. A = {(1,2,3), (1,-1,0)} II. B = {(1,1,0

Question

Consider the following subsets of R^3 I. A = {(1,2,3), (1,-1,0)} II. B = {(1,1,0), (1,2,3), (0,1,0)} III. C = {(1,1,0), (1,2,3), (0,1,-1)} IV. D = {(1,1,0), (1,2,3), (0,0,1)} Find all sets of vectors which is linearly independent. Find all sets of vectors which forms a basis for R^3 Suppose [v]_D = [1 -2 3]. Find [v]_s where S = ((1,0,0), (0,1,0), (0,0,1)} Find [w]_D if w = [-1 -4 -4] Find the dimension of the subspace of R^4 spanned by the vectors {(1,1,0,0), (1,2, -2,1), (3,6, -5,4), (1,0,2,3)}

Explanation / Answer

We have [v]D = (1,-2,3)T so that v= 1(1,1,0) -2 (1,2,3) +3(0,0,1) = (1,1,0)+ (-2,-4,-6) + (0,0,3) = (-1, -3,-3).

Then, v = -1(1,0,0) -3(0,1,0)-3(0,0,1) so that [v]s = ( -1 , -3 , -3 )T

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