Question

Let \({v_1} = \left( {\begin{array}{*{20}{c}}0\\0\\{ - 2}\end{array}} \right),{v_2} = \left( {\begin{array}{*{20}{c}}0\\{ - 3}\\8\end{array}} \right),{v_3} = \left( {\begin{array}{*{20}{c}}4\\{ - 1}\\{ - 5}\end{array}} \right)\) .

Does \(\left\{ {{v_1},{v_2},{v_3}} \right\}\) span \({R^3}\) ? Why or why not?

Short answer

\(\left\{ {{v_1},{v_2},{v_3}} \right\}\) are in span \({R^3}\).

Step-by-step solution

Reduce the matrix

First, reduce the row matrix \(\left\{ {{v_1},{v_2},{v_3}} \right\}\)to check whether it contains a pivot in each row.

So, the matrix \(\left\{ {{v_1},{v_2},{v_3}} \right\}\)can be written as:

\(\left\{ {{v_1},{v_2},{v_3}} \right\} = \left[ {\begin{array}{*{20}{c}}0&0&4\\0&{ - 3}&{ - 1}\\{ - 2}&8&{ - 5}\end{array}} \right]\)

Operations in rows

Apply row operation\({R_3} \leftrightarrow {R_1}\) in the given matrix.

\(\left\{ {{v_1},{v_2},{v_3}} \right\} = \left[ {\begin{array}{*{20}{c}}{ - 2}&8&{ - 5}\\0&{ - 3}&{ - 1}\\0&0&4\end{array}} \right]\)

Resultant matrix

The above matrix in terms of pivot can be written as:

\(\left\{ {{v_1},{v_2},{v_3}} \right\} = \left[ {\begin{array}{*{20}{c}} {\boxed{ - 2}}&8&{ - 5} \\ 0&{\boxed{ - 3}}&{ - 1} \\ 0&0&{\boxed4} \end{array}} \right]\)

Since the matrix \(\left\{ {{v_1},{v_2},{v_3}} \right\}\)has a pivot in each of the rows. As a result, span \({R^3}\) appears in the matrix column.

Hence, \(\left\{ {{v_1},{v_2},{v_3}} \right\}\)are in span \({R^3}\).