6: Let \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}1\\{ - 2}\\4\\3\end{array}} \right]\), \({{\mathop{\rm v}\nolimits} _2} = \left[ {\begin{array}{*{20}{c}}4\\{ - 7}\\9\\7\end{array}} \right]\), \[{{\mathop{\rm v}\nolimits} _3} = \left[ {\begin{array}{*{20}{c}}5\\{ - 8}\\6\\5\end{array}} \right]\], and \[{\mathop{\rm u}\nolimits} = \left[ {\begin{array}{*{20}{c}}{ - 4}\\{10}\\{ - 7}\\{ - 5}\end{array}} \right]\] . Determine if u is in the subspace of \({\mathbb{R}^4}\) generated by \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\).

Short Answer

Expert verified

\[{\mathop{\rm u}\nolimits} \] is not in the subspace of \({\mathbb{R}^4}\) generated by vectors \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\).

Step by step solution

01

Write the vector equation as an augmented matrix

When the vector equation \({{\mathop{\rm x}\nolimits} _1}{{\mathop{\rm v}\nolimits} _1} + {{\mathop{\rm x}\nolimits} _2}{{\mathop{\rm v}\nolimits} _2} + {{\mathop{\rm x}\nolimits} _3}{{\mathop{\rm v}\nolimits} _3} = {\mathop{\rm u}\nolimits} \) isconsistent, vector u is in thesubspace generated by vectors \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\).

The augmented matrix of the vector equation \({{\mathop{\rm x}\nolimits} _1}{{\mathop{\rm v}\nolimits} _1} + {{\mathop{\rm x}\nolimits} _2}{{\mathop{\rm v}\nolimits} _2} + {{\mathop{\rm x}\nolimits} _3}{{\mathop{\rm v}\nolimits} _3} = {\mathop{\rm u}\nolimits} \) is shown below:

\(\left[ {\begin{array}{*{20}{c}}{{{\mathop{\rm v}\nolimits} _1}}&{{{\mathop{\rm v}\nolimits} _2}}&{{{\mathop{\rm v}\nolimits} _3}}&{\mathop{\rm u}\nolimits} \end{array}} \right] \sim \left[ {\begin{array}{*{20}{c}}1&4&5&{ - 4}\\{ - 2}&{ - 7}&{ - 8}&{10}\\4&9&6&{ - 7}\\3&7&5&{ - 5}\end{array}} \right]\)

02

Apply the row operation

The following row operation demonstrates that u is not in the subspace generated by vectors \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\).

At row two, multiply row one by 2 and add it to row two. At row three, multiply row one by 4 and subtract it from row three. At row four, multiply row one by 3 and subtract it from row four.

\( \sim \left[ {\begin{array}{*{20}{c}}1&4&5&{ - 4}\\0&1&2&2\\0&{ - 7}&{ - 14}&9\\0&{ - 5}&{ - 10}&7\end{array}} \right]\)

At row three, multiply row two by 7 and add it to row three. At row four, multiply row two by 5 and add it to row four.

\( \sim \left[ {\begin{array}{*{20}{c}}1&4&5&{ - 4}\\0&1&2&2\\0&0&0&{23}\\0&0&0&{17}\end{array}} \right]\)

03

Determine whether u is in subspace generated by \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\)

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

.

The augmented matrix represents an inconsistent system. Therefore, \({\mathop{\rm u}\nolimits} \) is not in the subspace of \({\mathbb{R}^4}\) generated by vectors \(\left\{ {{{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}} \right\}\).

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Most popular questions from this chapter

A useful way to test new ideas in matrix algebra, or to make conjectures, is to make calculations with matrices selected at random. Checking a property for a few matrices does not prove that the property holds in general, but it makes the property more believable. Also, if the property is actually false, you may discover this when you make a few calculations.

38. Use at least three pairs of random \(4 \times 4\) matrices Aand Bto test the equalities \({\left( {A + B} \right)^T} = {A^T} + {B^T}\) and \({\left( {AB} \right)^T} = {A^T}{B^T}\). (See Exercise 37.) Report your conclusions. (Note:Most matrix programs use \(A'\) for \({A^{\bf{T}}}\).

Show that block upper triangular matrix \(A\) in Example 5is invertible if and only if both \({A_{{\bf{11}}}}\) and \({A_{{\bf{12}}}}\) are invertible. [Hint: If \({A_{{\bf{11}}}}\) and \({A_{{\bf{12}}}}\) are invertible, the formula for \({A^{ - {\bf{1}}}}\) given in Example 5 actually works as the inverse of \(A\).] This fact about \(A\) is an important part of several computer algorithims that estimates eigenvalues of matrices. Eigenvalues are discussed in chapter 5.

In Exercises 1–9, assume that the matrices are partitioned conformably for block multiplication. Compute the products shown in Exercises 1–4.

3. \[\left[ {\begin{array}{*{20}{c}}{\bf{0}}&I\\I&{\bf{0}}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}W&X\\Y&Z\end{array}} \right]\]

In exercises 11 and 12, mark each statement True or False. Justify each answer.

a. The definition of the matrix-vector product \(A{\bf{x}}\) is a special case of block multiplication.

b. If \({A_{\bf{1}}}\), \({A_{\bf{2}}}\), \({B_{\bf{1}}}\), and \({B_{\bf{2}}}\) are \(n \times n\) matrices, \[A = \left[ {\begin{array}{*{20}{c}}{{A_{\bf{1}}}}\\{{A_{\bf{2}}}}\end{array}} \right]\] and \(B = \left[ {\begin{array}{*{20}{c}}{{B_{\bf{1}}}}&{{B_{\bf{2}}}}\end{array}} \right]\), then the product \(BA\) is defined, but \(AB\) is not.

Generalize the idea of Exercise 21(a) [not 21(b)] by constructing a \(5 \times 5\) matrix \(M = \left[ {\begin{array}{*{20}{c}}A&0\\C&D\end{array}} \right]\) such that \({M^2} = I\). Make C a nonzero \(2 \times 3\) matrix. Show that your construction works.

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