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.

False, counterexample below

Step-by-step explanation:

Denote the unit vectors of R^3 by . Now consider and . We have that . Also, the vector is not a linear combination of 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 is not linearly independent, because is a non-trivial linear combination of these vectors that equals zero.

7

step-by-step explanation:

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step-by-step explanation: