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.)

35. If \({{\mathop{\rm v}\nolimits} _1}\) and \({v_2}\) are in \({\mathbb{R}^4}\) and \({{\mathop{\rm v}\nolimits} _2}\) is not a scalar multiple of \({v_1}\), then \(\left\{ {{v_1},{v_2}} \right\}\) is linearly independent.

Short answer

The given statement is false.

Step-by-step solution

Determine whether the given statement is true or false

If set \(S = \left\{ {{{\mathop{\rm v}\nolimits} _1},...,{v_p}} \right\}\) in \({\mathbb{R}^n}\) contains the zero vector, then it islinearly dependent.

If the vector is a zero vector, then it is not linearly independent.

Thus, the given statement is false.

Construct an example to show that the statement is not always true

Take \({{\mathop{\rm v}\nolimits} _1}\) to be not a scalar multiple of vector \({{\mathop{\rm v}\nolimits} _2}\).

\({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}0\\0\\0\\0\end{array}} \right]\), \({{\mathop{\rm v}\nolimits} _2} = \left[ {\begin{array}{*{20}{c}}1\\1\\1\\1\end{array}} \right]\)

Since the vector in \({\mathbb{R}^4}\) contains the zero vector, the set is not linearly independent.

Thus, the given statement is false.