Question

his is LINEAR ALGEBRA & MATRICIES Ill give thumbs up for steps on how to do the problem and solutions.

your responses. (a) The dimension of Maa l (b) Does S span the vector space Maxa (c) Is S a basis for M2zxa? Justify your answer.

Answer 1

s = {[1 0 1 0], [0 1 0 1] , [1 0 0 1],[0 1 1 1]} a) The dimension of n_2 x_2 is 4 b) let [a b c d] = x[1 0 1 0] + beta[0 1 0 1] + gamma[1 0 0 1] + Gamma[0 1 1 1] = [alpha 0 alpha 1] + [0 beta 0 beta] + [gamma 0 0 gamma] + [a] = alpha + gamma alpha + beta + Beta + gamma +] = > alpha + gamma = a, alpha + = b , beta + = c beta + gamma + = d = > gamma + c = d = > r = c - d = alpha = a - c + d beta = c - b + a - c + d beta = -b + a + d Here s space (R) c) s is a basis for m_2 times 2 (r) as dim (m_2 times 2(R)) = 4 and s has exactly A vectors thus any spanning set of m_2 times 2 (R) having 4 elements is a basis for m_2 times 2 (R)

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