Question

Let \(b = \left[ {\begin{array}{*{20}{c}}{ - 1}\\1\\0\end{array}} \right]\), and let \(A\) be the matrix in exercise 9. Is \(b\)in the range of linear transformation\(x \mapsto Ax\)? Why or why not?

Short answer

The system represented by \(\left[ {A\,\,b} \right]\) is consistent; so \(b\) is in the range of \(x \to Ax\).

Step-by-step solution

Formation of the augmented matrix

Using matrix \(A = \left[ {\begin{array}{*{20}{c}}1&{ - 4}&7&{ - 5}\\0&1&{ - 4}&3\\2&{ - 6}&6&{ - 4}\end{array}} \right]\) and \(b = \left[ {\begin{array}{*{20}{c}}{ - 1}\\1\\0\end{array}} \right]\), form the augmented matrix \(\left[ {A\,\,b} \right]\).

\(\left[ {\begin{array}{*{20}{c}}1&{ - 4}&7&{ - 5}&{ - 1}\\0&1&{ - 4}&3&1\\2&{ - 6}&6&{ - 4}&0\end{array}} \right]\)

Simplification of the augmented matrix using row operations

Simplify the augmented matrix using row operations.

At row 2, multiply row 1 with 2 and subtract it from row 3, i.e. \({R_3} \to {R_3} - 2{R_1}\).

\(\left[ {\begin{array}{*{20}{c}}1&{ - 4}&7&{ - 5}&{ - 1}\\0&1&{ - 4}&3&1\\{2 - 2\left( 1 \right)}&{ - 6 - 2\left( { - 4} \right)}&{6 - 2\left( 7 \right)}&{ - 4 - 2\left( { - 5} \right)}&{0 - 2\left( { - 1} \right)}\end{array}} \right]\)

After the row operation, the matrix will become:

\(\left[ {\begin{array}{*{20}{c}}1&{ - 4}&7&{ - 5}&{ - 1}\\0&1&{ - 4}&3&1\\0&2&{ - 8}&6&2\end{array}} \right]\)

Simplification of the augmented matrix using row operations

At row 3, multiply row 2 with 2 and subtract it from row 3, i.e., \({R_3} \to {R_3} - 2{R_2}\).

\(\left[ {\begin{array}{*{20}{c}}1&{ - 4}&7&{ - 5}&{ - 1}\\0&1&{ - 4}&3&1\\0&{2 - 2\left( 1 \right)}&{ - 8 - 2\left( { - 4} \right)}&{6 - 2\left( 3 \right)}&{2 - 2\left( 1 \right)}\end{array}} \right]\)

After the row operation, the matrix will become:

\(\left[ {\begin{array}{*{20}{c}}1&{ - 4}&7&{ - 5}&{ - 1}\\0&1&{ - 4}&3&1\\0&0&0&0&0\end{array}} \right]\)

As the system given by \(\left[ {A\,\,b} \right]\) is consistent, so \(b\) is in the range of transformation \(x \to Ax\).