Question

The following statement is either true​ (in all​ cases) or false​ (for at least one​ example). If​ false, construct a specific example to show that the statement is not always true. Such an example is called a counterexample to the statement. If a statement is​ true, give a justification. 1. If v_1, v_2, v_3 are in R^3 and v_3 is not a linear combination of v_1, v_2, then {v_1, v_2, v_3} is linearly independent.

Answer 1

Answer:

False, counterexample below

Step-by-step explanation:

Denote the unit vectors of R^3 by $e_1=(1,0,0), e_2=(0,1,0), e_3=(0,0,1)$. Now consider $v_1=e_1, v_2=2e_1$ and $v_3=e_3$. We have that $v_1, v_2,v_3 \in \mathbb{R}^3$. Also, the vector $v_3$ is not a linear combination of $v_1, v_2$ because any linear combination of these two vectors will have third coordinate zero, but v_3 has third coordinate 1 so they can't be equal.

However, the set $\{v_1, v_2,v_3\}$ is not linearly independent, because $2v_1-v_2+0v_3=0$ is a non-trivial linear combination of these vectors that equals zero.