Question

Exercise 31 reveal an important connection between linear independence and linear transformations and provide practice using the definition of linear dependence. Let V and W be vector spaces, let \(T:V \to W\) be a linear transformation, and let \(\left\{ {{v_{\bf{1}}},...,{v_p}} \right\}\) be a subset of V.

31. Show that if \(\left\{ {{v_{\bf{1}}},...,{v_p}} \right\}\) is linearly dependent in V, then the set of images, \(\left\{ {T\left( {{v_{\bf{1}}}} \right),...,T\left( {{v_p}} \right)} \right\}\), is linearly dependent in W. This fact shows that if a linear transformation maps a set \(\left\{ {{v_{\bf{1}}},...,{v_p}} \right\}\) onto a linearly independent set \(\left\{ {T\left( {{v_{\bf{1}}}} \right),...,T\left( {{v_p}} \right)} \right\}\), then the original set is linearly independent, too (because it cannot be linearly dependent).

Short answer

The set \(\left\{ {T\left( {{v_1}} \right),...,T\left( {{v_p}} \right)} \right\}\) is linearly dependent.

Step-by-step solution

Write the given statement

Suppose \(\left\{ {{v_1},...,{v_p}} \right\}\) is linearly dependent.

Use the definition of linear dependence

There exist scalars \({c_1},...,{c_p}\) that are not all zeros, such that

\({c_1}{v_1} + ... + {c_p}{v_p} = 0\).

Take T on both sides.

\(\begin{array}{c}T\left( {{c_1}{v_1} + ... + {c_p}{v_p}} \right) = T\left( 0 \right)\\{c_1}T\left( {{v_1}} \right) + ... + {c_p}T\left( {{v_p}} \right) = 0\end{array}\)

(Since not all \({c_i}\)s are zeros)

Draw a conclusion

Thus, \(\left\{ {T\left( {{v_1}} \right),...,T\left( {{v_p}} \right)} \right\}\) is linearly dependent.