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Chapter 2: Q2.8-5Q (page 93)

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

Short Answer

Expert verified

\[{\mathop{\rm w}\nolimits} \] is not in the subspace of \({\mathbb{R}^3}\) generated by vectors \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\).

Step by step solution

01

Write the vector equation as an augmented matrix

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

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

\(\left[ {\begin{array}{*{20}{c}}{{{\mathop{\rm v}\nolimits} _1}}&{{{\mathop{\rm v}\nolimits} _2}}&{\mathop{\rm w}\nolimits} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}}2&{ - 4}&8\\3&{ - 5}&2\\{ - 5}&8&{ - 9}\end{array}} \right]\)

02

Apply the row operation

The following row operation demonstrates that w is not in the subspace generated by vectors \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\).

At row one, multiply row one by \(\frac{1}{2}\).

\( \sim \left[ {\begin{array}{*{20}{c}}1&{ - 2}&4\\3&{ - 5}&2\\{ - 5}&8&{ - 9}\end{array}} \right]\)

At row two, multiply row one by \(3\) and subtract it from row two. At row three, multiply row one by 5 and add it to row three.

\( \sim \left[ {\begin{array}{*{20}{c}}1&{ - 2}&4\\0&1&{ - 10}\\0&{ - 2}&{11}\end{array}} \right]\)

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

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

03

Determine whether w is in subspace generated by \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\)

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

.

The augmented matrix represents an inconsistent system. Therefore, w is not in the subspace of \({\mathbb{R}^3}\) generated by vectors \({{\mathop{\rm v}\nolimits} _1}\) and \({{\mathop{\rm v}\nolimits} _2}\).

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Most popular questions from this chapter

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

4. \[\left[ {\begin{array}{*{20}{c}}I&0\\{ - X}&I\end{array}} \right]\left[ {\begin{array}{*{20}{c}}A&B\\C&D\end{array}} \right]\]

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.

36. Write the command(s) that will create a \(6 \times 4\) matrix with random entries. In what range of numbers do the entries lie? Tell how to create a \(3 \times 3\) matrix with random integer entries between \( - {\bf{9}}\) and 9. (Hint:If xis a random number such that 0 < x < 1, then \( - 9.5 < 19\left( {x - .5} \right) < 9.5\).

Let \(T:{\mathbb{R}^n} \to {\mathbb{R}^n}\) be an invertible linear transformation. Explain why T is both one-to-one and onto \({\mathbb{R}^n}\). Use equations (1) and (2). Then give a second explanation using one or more theorems.

In Exercises 1 and 2, compute each matrix sum or product if it is defined. If an expression is undefined, explain why. Let

\(A = \left( {\begin{aligned}{*{20}{c}}2&0&{ - 1}\\4&{ - 5}&2\end{aligned}} \right)\), \(B = \left( {\begin{aligned}{*{20}{c}}7&{ - 5}&1\\1&{ - 4}&{ - 3}\end{aligned}} \right)\), \(C = \left( {\begin{aligned}{*{20}{c}}1&2\\{ - 2}&1\end{aligned}} \right)\), \(D = \left( {\begin{aligned}{*{20}{c}}3&5\\{ - 1}&4\end{aligned}} \right)\) and \(E = \left( {\begin{aligned}{*{20}{c}}{ - 5}\\3\end{aligned}} \right)\)

\( - 2A\), \(B - 2A\), \(AC\), \(CD\).

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.
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