Question: Let \(u = \left( {\begin{array}{*{20}{c}}2\\{ - 3}\\2\end{array}} \right)\) and \(A = \left( {\begin{array}{*{20}{c}}5&8&7\\0&1&{ - 1}\\1&3&0\end{array}} \right)\). Is \(u\) in the subset of \({\mathbb{R}^3}\) spanned by the columns of \(A\)? Why or why not?

The matrix equation \(Ax = b\) has the same solution set as the vector equation \({x_1}{a_1} + {x_2}{a_2} + ... + {x_n}{a_n} = b\) has the same solution set as the system of linear equations whoseaugmented matrix is\(\left( {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{...}&{{a_n}}&b\end{array}} \right)\).

Write the given matrix in the augmented form \(\left( {\begin{array}{*{20}{c}}A&u\end{array}} \right)\).

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

Perform an elementary row operationto produce the first augmented matrix.

Interchange rowsone and three.

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

Perform an elementary row operationto produce the second augmented matrix.

Write the sum of \( - 5\) times row one and row three in row three.

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

Perform an elementary row operationto produce the third augmented matrix.

Write the sum of \(7\) times row two and row three in row three.

\(\left( {\begin{array}{*{20}{c}}1&3&0&2\\0&1&{ - 1}&{ - 3}\\0&0&0&{ - 29}\end{array}} \right)\)

To obtain the solution of the vector equation, convert the augmented matrix into vector equations.

Write the obtained matrix,\(\left( {\begin{array}{*{20}{c}}1&1&4\\0&8&{12}\\0&0&0\end{array}} \right)\),inthe equation notation.

\(\begin{array}{c}{x_1} + {x_2} = 4\\8{x_2} = 12\end{array}\)

Every \(b\) in \({\mathbb{R}^m}\) is a linear combination of the columns of \(A\). A set of vectors \(\left\{ {{v_1},...,{v_p}} \right\}\)in \({\mathbb{R}^m}\) spans \({\mathbb{R}^m}\) if every vector in \({\mathbb{R}^m}\) is a linear combination of \({v_1},...,{v_p}\). The equation \(Ax = b\) has a solution if and only if \(b\) is a linear combination of the columns of \(A\).

There is no solution to the equation \(Ax = u\).

Hence, \(u\) is not in the subset \({\mathbb{R}^3}\) spanned by the columns of A.