Question

Let V be a vector space that contains a linearly independent set \(\left\{ {{u_{\bf{1}}},{u_{\bf{2}}},{u_{\bf{3}}},{u_{\bf{4}}}} \right\}\). Describe how to construct a set of vectors \(\left\{ {{v_{\bf{1}}},{v_{\bf{2}}},{v_{\bf{3}}},{v_{\bf{4}}}} \right\}\) in V such that \(\left\{ {{v_{\bf{1}}},{v_{\bf{3}}}} \right\}\) is a basis for Span\(\left\{ {{v_{\bf{1}}},{v_{\bf{2}}},{v_{\bf{3}}},{v_{\bf{4}}}} \right\}\).

Short answer

If \({v_1}\) and \({v_3}\) are linearly independent and \({v_2} = a{v_1} + b{v_3}\) and \({v_4} = c{v_1} + d{v_3}\), wherea, b, c, and d are scalar, then \(\left\{ {{v_1},{v_3}} \right\}\) is a basis for \({\rm{Span}}\left\{ {{v_1},{v_2},{v_3},{v_4}} \right\}\).

Step-by-step solution

Construct the linearly independent vectors

Let \({v_1}\) and \({v_3}\) be linearly independent.

Construct a relation between the vectors, except \({v_{\bf{1}}}\) and

Let \({v_2} = a{v_1} + b{v_3}\) and \({v_4} = c{v_1} + d{v_3}\), wherea, b, c, and d are scalar.

By the spanning set theorem, \({\rm{Span}}\left\{ {{v_1},{v_2},{v_3},{v_4}} \right\} = {\rm{Span}}\left\{ {{v_1},{v_3}} \right\}\).

Draw a conclusion

In this way, \(\left\{ {{v_1},{v_3}} \right\}\) forms a basis for \({\rm{Span}}\left\{ {{v_1},{v_2},{v_3},{v_4}} \right\}\).