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.

Step by step solution

01

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.

02

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.

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