Question 24: Mark each statement True or False. Justify each answer.

a. Every matrix equation\(Ax = b\)corresponds to a vector equation with the same solution set.

b. Any linear combination of vectors can always be written in the form\(Ax\)for a suitable matrix A and vector\(x\).

c. The solution set of a linear system whose augmented matrix is\(\left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}&b\end{array}} \right]\)is the same as the solution set of\(Ax = b\), if\(A = \left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}\end{array}} \right]\).

d. If the equation\(Ax = b\)is inconsistent, then b is not in the set spanned by the columns of A.

e. If the augmented matrix\(\left[ {\begin{array}{*{20}{c}}A&b\end{array}} \right]\)has a pivot position in every row, then the equation\(Ax = b\)is inconsistent.

f. If A is an \(m \times n\) matrix whose columns do not span \({R^m}\), then the equation \(Ax = b\) is inconsistent for some \(b\) in \({R^m}\).

If A is a \(m \times n\) matrix with columns \({a_1},{a_2},........,{a_n}\) and b is in \({R^m}\), the matrix equation \(Ax = b\) has the same solution set as the vector equation \({a_1}{x_1} + {a_2}{x_2} + {a_3}{x_3} + ........{a_n}{x_n} = b\), which in turn has the same solution set as the system of linear equations whose augmented matrix is \(\left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}&{.......}&{{a_n}}&b\end{array}} \right]\).

Hence, the given statement is true.

For a suitable matrix A and vector \(x\), any linear combination of vectors can always be represented in the form \(Ax\).

For example, for the linear combination of \(2{v_1} - 3{v_2} + 6{v_3}\), the matrix times a vector can be written as:

\(2{v_1} - 3{v_2} + 6{v_3} = \left[ {\begin{array}{*{20}{c}}{{v_1}}&{{v_2}}&{{v_3}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}2\\{ - 3}\\6\end{array}} \right]\)

Hence, the given statement is true.

The matrix equation \(Ax = b\) has the same solution set as the vector equation, which in turn has the same solution set as the system of linear equations whose augmented matrix is \(\left[ {\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}&{.......}&{{a_n}}&b\end{array}} \right]\), if A is a \(m \times n\) matrix with columns \({a_1},{a_2},........,{a_n}\) and b is in \({R^m}\).

Hence, the given statement is true.

We know that if \(b\) is not in the set covered by A's columns, then it is correct to assert that \(b\) is not a linear combination of A's columns.

Hence, the given statement is true.

Recall that if every row of an augmented matrix \(\left[ {\begin{array}{*{20}{c}}A&b\end{array}} \right]\) has a pivot position, the equation \(Ax = b\) may or may not be consistent.

Hence, the given statement is false.

Assume A is a \(m \times n\) matrix. Then,

a. The equation \(Ax = b\) has a solution for each b in \({R^m}\).

b. \({R^m}\) is spanned by the columns of A.

Hence, the given statement is true.