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Question: determine if the statement is true or false and justify...

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Determine if the statement is true or false, and justify your answer.

If u4 is not a linear combination of {u1, u2, u3}, then {u1, u2, u3, u4} is linearly independent.

A) False. Consider u1 = (1, 0, 0), u2 = (0, 1, 0), u3 = (0, 0, 1), u4 = (0, 1, 0).

B) False. Consider u1 = (1, 0, 0), u2 = (1, 0, 0), u3 = (1, 0, 0), u4 = (0, 1, 0).

C) True. The echelon form of the augmented matrix [u1u2u3u4] will have at least one row of zeroes at the bottom, which means the vectors are linearly independent.

D) False. The echelon form of the augmented matrix [u1u2u3u4] will have at least one row of zeroes at the bottom, which means the vectors are linearly dependent.

E) True. If {u1, u2, u3, u4} is linearly dependent, then u4 = x1u1 + x2u2 + x3u3, which is a contradiction.

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