Question

4. (20 points) Determine whether the following sets are linear independent. Prove your statem ent. in the vector space of 2 × 2 matrices. b. (sinx, cos.x, 1 in the vector space CTo, 1].

Answer 1

Consider s = {(1 1 1 0), (2 1 2 0), (1 0 1 0)} s = {v_1, v_2, v_3} where v_1 = (1 1 1 0)_2 times 2, v_2 = (2 1 2 0)+ 2 times 2, v_3 = (1 0 1 0)_2 times 2 we can be written as linear combination the given vector (matrices) v_1, v_2, v_3 v_2 alpha v_1, beta v_2 (2 1 2 0) = alpha (1 1 1 0) + beta (1 0 1 0) if alpha = 1 and beta = 1 (1 1 2 0) = 1 (1 1 1 0) + 1 (1 0 1 0) (2 1 2 0) = (1 1 1 0) + (1 0 1 0) (2 1 2 0) = (2 1 2 0) Hence s = {(1 1 1 0), (2 1 2 0), (1 0 1 0)} linearly dependent set Hence this set s is not vector of 2 times 2 matrix Because v_2 written v_1 and v_3 from \r\n {sin alpha }