7: Let \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}2\\{ - 8}\\6\end{array}} \right]\), \({{\mathop{\rm v}\nolimits} _2} = \left[ {\begin{array}{*{20}{c}}{ - 3}\\8\\{ - 7}\end{array}} \right]\), \({{\mathop{\rm v}\nolimits} _3} = \left[ {\begin{array}{*{20}{c}}{ - 4}\\6\\{ - 7}\end{array}} \right]\), \({\mathop{\rm p}\nolimits} = \left[ {\begin{array}{*{20}{c}}6\\{ - 10}\\{11}\end{array}} \right]\), and\(A = \left[ {\begin{array}{*{20}{c}}{{{\mathop{\rm v}\nolimits} _1}}&{{{\mathop{\rm v}\nolimits} _2}}&{{{\mathop{\rm v}\nolimits} _3}}\end{array}} \right]\).

1. How many vectors are in \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\)?
2. How many vectors are in Col A?
3. Is p in Col A? Why or why not?

(a)

The set \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\)contains three vectors: \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},\) and \({{\mathop{\rm v}\nolimits} _3}\).

Span\(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\} = {\mathop{\rm Col}\nolimits} \,\,A\) contains an infinite number of vectors.

(c)

Vector p is alinear combination of the columns of A if and only if b can be written as Ax for some x, that is, if and only if the equation \(Ax = b\) has asolution.

Write matrix A in the form \(\left[ {\begin{array}{*{20}{c}}{{{\mathop{\rm v}\nolimits} _1}}&{{{\mathop{\rm v}\nolimits} _2}}&{{{\mathop{\rm v}\nolimits} _3}}\end{array}} \right]\), as shown below:

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

The augmented matrix \(\left[ {\begin{array}{*{20}{c}}A&{\mathop{\rm p}\nolimits} \end{array}} \right]\) is shown below:

\(\left[ {\begin{array}{*{20}{c}}A&{\mathop{\rm p}\nolimits} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}}2&{ - 3}&{ - 4}&6\\{ - 8}&8&6&{ - 10}\\6&{ - 7}&{ - 7}&{11}\end{array}} \right]\)

At row two, multiply row one by 4 and add it to row two. At row three, multiply row one by 3 and subtract it from row three.

\[ \sim \left[ {\begin{array}{*{20}{c}}2&{ - 3}&{ - 4}&6\\0&{ - 4}&{ - 10}&{14}\\0&2&5&{ - 7}\end{array}} \right]\]

At row three, multiply row three by 2 and add it to row two.

\[ \sim \left[ {\begin{array}{*{20}{c}}2&{ - 3}&{ - 4}&6\\0&{ - 4}&{ - 10}&{14}\\0&0&0&0\end{array}} \right]\]

Thecolumn spaceof matrix A is the set Col A of all linear combinationsof the columns of A.

Mark the pivot column in the row echelon form of the matrix

.

Since the equation \(A{\mathop{\rm x}\nolimits} = p\) has a solution, p is in Col A.