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# Determine whether given statements about independent vectors are true or false

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

1. The vectors

are linearly independent.

2. The vectors

are linearly independent.

3. The vectors

are linearly independent.

1. Proof. Consider the equation

But, 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

2. Proof. Consider the equation

Since are linearly independent (by part (a)) we have

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

3. Proof. Consider the equation

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