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 counterexampleto 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.)

37. If \({{\bf{v}}_1},...,{{\bf{v}}_4}\) are in \({\mathbb{R}^4}\) and \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}\) is linearly dependent, then \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3},{{\bf{v}}_4}} \right\}\) is also linearly dependent.

Short answer

The statement is true.

Step-by-step solution

Write the condition for the set to be linearly dependent

The vectors are said to be linearly dependent if the equation is in the form of \({c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} + ... + {c_p}{{\bf{v}}_p} = 0\), where \({c_1},{c_2},...,{c_p}\) are scalars.

Check whether the statement is true or false

It is given that \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}\)are linearly dependent, so \({{\bf{v}}_3}\) is the linear combinations of vectors \({{\bf{v}}_1}\) and \({{\bf{v}}_2}\).

It shows that if any vector from the set \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3}} \right\}\) is a linear combination of other vectors, then \(\left\{ {{{\bf{v}}_1},{{\bf{v}}_2},{{\bf{v}}_3},{{\bf{v}}_4}} \right\}\) would also be in the linear combination.

Thus, the statement is true.