Question

Each statement in Exercises 33-38 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. (One specific example cannot explain why a statement is always true. You will have to do more work here than in Exercises 21 and 22.)

33. If \({{\mathop{\rm v}\nolimits} _1},...,{v_4}\) are in \({\mathbb{R}^4}\) and \({{\mathop{\rm v}\nolimits} _3} = 2{{\mathop{\rm v}\nolimits} _1} + {v_2}\), then \(\left\{ {{v_1},{v_2},{v_3},{v_4}} \right\}\) is linearly dependent.

Short answer

The given statement is true.

Step-by-step solution

Determine whether the given statement is true or false

An indexed set \(S = \left\{ {{{\mathop{\rm v}\nolimits} _1},...,{v_p}} \right\}\) of two or more vectors islinearly dependentif and only if one of the vectors in \(S\) is a linear combination of the others.

Thus, the given statement is true.

Explain why the given statement is true

The given linear combination of vectors in \({\mathbb{R}^4}\) is \({{\mathop{\rm v}\nolimits} _3} = 2{{\mathop{\rm v}\nolimits} _1} + {v_2}\).

Since one of the vectors in \({\mathbb{R}^4}\) is a linear combination of the others, the vectors in \({\mathbb{R}^4}\) are linearly dependent.

Thus, the given statement is true.