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Chapter 4: Q17E (page 191)

Question:In Exercises 15鈥�18, find a basis for the space spanned by the given vectors,\({{\bf{v}}_{\bf{1}}}, \ldots ,{{\bf{v}}_{\bf{5}}}\).

17. \(\left( {\begin{array}{*{20}{c}}8\\9\\{ - 3}\\{ - 6}\\0\end{array}} \right)\), \(\left( {\begin{array}{*{20}{c}}{\bf{4}}\\{\bf{5}}\\{\bf{1}}\\{ - {\bf{4}}}\\{\bf{4}}\end{array}} \right)\), \(\left( {\begin{array}{*{20}{c}}{ - {\bf{1}}}\\{ - {\bf{4}}}\\{ - {\bf{9}}}\\{\bf{6}}\\{ - {\bf{7}}}\end{array}} \right)\), \(\left( {\begin{array}{*{20}{c}}{\bf{6}}\\{\bf{8}}\\{\bf{4}}\\{ - {\bf{7}}}\\{{\bf{10}}}\end{array}} \right)\), \(\left( {\begin{array}{*{20}{c}}{ - {\bf{1}}}\\{\bf{4}}\\{{\bf{11}}}\\{ - {\bf{8}}}\\{ - {\bf{7}}}\end{array}} \right)\)

Short Answer

Expert verified

The basis for the space spanned by the vectors is \(\left\{ {\left( {\begin{array}{*{20}{c}}8\\9\\{ - 3}\\{ - 6}\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}4\\5\\1\\{ - 4}\\4\end{array}} \right),\left( {\begin{array}{*{20}{c}}{ - 1}\\{ - 4}\\{ - 9}\\6\\{ - 7}\end{array}} \right)} \right\}\).

Step by step solution

01

State the basis for Col A

The set of all linear combinations of the columns of matrix A is Col A.It is called the column space of A. Pivotcolumns are the basis for Col A.

02

Obtain the row-reduced echelon form

Consider the vectors\(\left( {\begin{array}{*{20}{c}}8\\9\\{ - 3}\\{ - 6}\\0\end{array}} \right)\),\(\left( {\begin{array}{*{20}{c}}4\\5\\1\\{ - 4}\\4\end{array}} \right)\),\(\left( {\begin{array}{*{20}{c}}{ - 1}\\{ - 4}\\{ - 9}\\6\\{ - 7}\end{array}} \right)\),\(\left( {\begin{array}{*{20}{c}}6\\8\\4\\{ - 7}\\{10}\end{array}} \right)\),\(\left( {\begin{array}{*{20}{c}}{ - 1}\\4\\{11}\\{ - 8}\\{ - 7}\end{array}} \right)\).

Five vectors span the column spaceof a matrix. So, construct matrix A by using the given vectors as shown below:

\(A = \left( {\begin{array}{*{20}{c}}8&4&{ - 1}&6&{ - 1}\\9&5&{ - 4}&8&4\\{ - 3}&1&{ - 9}&4&{11}\\{ - 6}&{ - 4}&6&{ - 7}&{ - 8}\\0&4&{ - 7}&{10}&{ - 7}\end{array}} \right)\)

Use the code in MATLAB to obtain the row-reduced echelon form as shown below:

\( > > {\rm{ U}} = {\rm{rref}}\left( {\rm{A}} \right)\)

\(\left( {\begin{array}{*{20}{c}}8&4&{ - 1}&6&{ - 1}\\9&5&{ - 4}&8&4\\{ - 3}&1&{ - 9}&4&{11}\\{ - 6}&{ - 4}&6&{ - 7}&{ - 8}\\0&4&{ - 7}&{10}&{ - 7}\end{array}} \right) \sim \left( {\begin{array}{*{20}{c}}1&0&0&{ - 1/2}&3\\0&1&0&{5/2}&{ - 7}\\0&0&1&0&{ - 3}\\0&0&0&0&0\\0&0&0&0&0\end{array}} \right)\)

03

Write the basis for Col A

To identify the pivot and the pivot position, observe the leftmost column (nonzero column) matrix, that is, the pivot column. At the top of this column, 1 is the pivot.

\(A = \left[ {\begin{array}{*{20}{c}} {\boxed1}&0&0&{ - \frac{1}{2}}&3 \\ 0&{\boxed1}&0&{\frac{5}{2}}&{ - 7} \\ 0&0&{\boxed1}&0&{ - 3} \\ 0&0&0&0&0 \\ 0&0&0&0&0 \end{array}} \right]\)

The first, second, and third columns have pivot elements.

The corresponding columns of matrix A are shown below:

\(\left( {\begin{array}{*{20}{c}}8\\9\\{ - 3}\\{ - 6}\\0\end{array}} \right)\),\(\left( {\begin{array}{*{20}{c}}4\\5\\1\\{ - 4}\\4\end{array}} \right)\),\(\left( {\begin{array}{*{20}{c}}{ - 1}\\{ - 4}\\{ - 9}\\6\\{ - 7}\end{array}} \right)\)

The column space is shown below:

\({\rm{Col }}A = \left\{ {\left( {\begin{array}{*{20}{c}}8\\9\\{ - 3}\\{ - 6}\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}4\\5\\1\\{ - 4}\\4\end{array}} \right),\left( {\begin{array}{*{20}{c}}{ - 1}\\{ - 4}\\{ - 9}\\6\\{ - 7}\end{array}} \right)} \right\}\)

Thus, the basis for Col Ais \(\left\{ {\left( {\begin{array}{*{20}{c}}8\\9\\{ - 3}\\{ - 6}\\0\end{array}} \right),\left( {\begin{array}{*{20}{c}}4\\5\\1\\{ - 4}\\4\end{array}} \right),\left( {\begin{array}{*{20}{c}}{ - 1}\\{ - 4}\\{ - 9}\\6\\{ - 7}\end{array}} \right)} \right\}\).

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Most popular questions from this chapter

Suppose \(A\) is \(m \times n\)and \(b\) is in \({\mathbb{R}^m}\). What has to be true about the two numbers rank \(\left[ {A\,\,\,{\rm{b}}} \right]\) and \({\rm{rank}}\,A\) in order for the equation \(Ax = b\) to be consistent?

Question: Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix \(A\) is \(m \times n\).

13. Show that if \(P\) is an invertible \(m \times m\) matrix, then rank\(PA\)=rank\(A\).(Hint: Apply Exercise12 to \(PA\) and \({P^{ - 1}}\left( {PA} \right)\).)

In Exercise 17, Ais an \(m \times n\] matrix. Mark each statement True or False. Justify each answer.

17. a. The row space of A is the same as the column space of \({A^T}\].

b. If B is any echelon form of A, and if B has three nonzero rows, then the first three rows of A form a basis for Row A.

c. The dimensions of the row space and the column space of A are the same, even if Ais not square.

d. The sum of the dimensions of the row space and the null space of A equals the number of rows in A.

e. On a computer, row operations can change the apparent rank of a matrix.

Define a linear transformation by \(T\left( {\mathop{\rm p}\nolimits} \right) = \left( {\begin{array}{*{20}{c}}{{\mathop{\rm p}\nolimits} \left( 0 \right)}\\{{\mathop{\rm p}\nolimits} \left( 0 \right)}\end{array}} \right)\). Find \(T:{{\mathop{\rm P}\nolimits} _2} \to {\mathbb{R}^2}\)polynomials \({{\mathop{\rm p}\nolimits} _1}\) and \({{\mathop{\rm p}\nolimits} _2}\) in \({{\mathop{\rm P}\nolimits} _2}\) that span the kernel of T, and describe the range of T.

Consider the polynomials \({{\bf{p}}_{\bf{1}}}\left( t \right) = {\bf{1}} + t\), \({{\bf{p}}_{\bf{2}}}\left( t \right) = {\bf{1}} - t\), \({{\bf{p}}_{\bf{3}}}\left( t \right) = {\bf{4}}\), \({{\bf{p}}_{\bf{4}}}\left( t \right) = {\bf{1}} + {t^{\bf{2}}}\), and \({{\bf{p}}_{\bf{5}}}\left( t \right) = {\bf{1}} + {\bf{2}}t + {t^{\bf{2}}}\), and let H be the subspace of \({P_{\bf{5}}}\) spanned by the set \(S = \left\{ {{{\bf{p}}_{\bf{1}}},\,{{\bf{p}}_{\bf{2}}},\;{{\bf{p}}_{\bf{3}}},\,{{\bf{p}}_{\bf{4}}},\,{{\bf{p}}_{\bf{5}}}} \right\}\). Use the method described in the proof of the Spanning Set Theorem (Section 4.3) to produce a basis for H. (Explain how to select appropriate members of S.)
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