Consider two linearly independent vectors . Determine whether each of the following statements is true or false.

- The vectors
are linearly independent.

- The vectors
are linearly independent.

- The vectors
are linearly independent.

*Proof.*Consider the equationBut, by Theorem 3.13 (page 484) we know that if and are independent then are independent. Hence, this second equation is true only if . But, implies that . Hence, we have , establishing the linear independence of

*Proof.*Consider the equationSince are linearly independent (by part (a)) we have

But these equations are only satisfied for . Hence, this establishes the linear independence of

*Proof.*Consider the equationBut from Theorem 13.13 (page 484) we know that if are independent, then are independent. Thus, we must have , establishing the independence of