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

False, counterexample below

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.