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

###### Question details

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

If **u**_{4} is *not* a linear
combination of {**u**_{1},
**u**_{2}, **u**_{3}},
then {**u**_{1},
**u**_{2}, **u**_{3},
**u**_{4}} is linearly independent.

A) False. Consider **u**_{1} = (1, 0, 0),
**u**_{2} = (0, 1, 0),
**u**_{3} = (0, 0, 1),
**u**_{4} = (0, 1, 0).

B) False. Consider **u**_{1} = (1, 0, 0),
**u**_{2} = (1, 0, 0),
**u**_{3} = (1, 0, 0),
**u**_{4} = (0, 1, 0).

C) True. The echelon form of the augmented matrix
[**u**_{1}**u**_{2}**u**_{3}**u**_{4}]
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
[**u**_{1}**u**_{2}**u**_{3}**u**_{4}]
will have at least one row of zeroes at the bottom, which means the
vectors are linearly dependent.

E) True. If {**u**_{1},
**u**_{2}, **u**_{3},
**u**_{4}} is linearly dependent, then
**u**_{4} =
x_{1}**u**_{1} +
x_{2}**u**_{2} +
x_{3}**u**_{3}, which is a
contradiction.