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The vectors \({{\mathop{\rm v}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}1\\{ - 3}\end{array}} \right],{{\mathop{\rm v}\nolimits} _2} = \left[ {\begin{array}{*{20}{c}}2\\{ - 8}\end{array}} \right],{{\mathop{\rm v}\nolimits} _3} = \left[ {\begin{array}{*{20}{c}}{ - 3}\\7\end{array}} \right]\) span \({\mathbb{R}^2}\) but do not form a basis. Find two different ways to express \(\left[ {\begin{array}{*{20}{c}}1\\1\end{array}} \right]\) as a linear combination of \({{\mathop{\rm v}\nolimits} _1},{{\mathop{\rm v}\nolimits} _2},{{\mathop{\rm v}\nolimits} _3}\).

Short Answer

Expert verified

The two different ways to express \(\left( {\begin{array}{*{20}{c}}1\\1\end{array}} \right)\) as a linear combination are \(5{{\mathop{\rm v}\nolimits} _1} - 2{{\mathop{\rm v}\nolimits} _2}\) and \(10{{\mathop{\rm v}\nolimits} _1} - 3{{\mathop{\rm v}\nolimits} _2} + {{\mathop{\rm v}\nolimits} _3}\).

Step by step solution

01

Write the vector equation as an augmented matrix

To solve the vector equation \({{\mathop{\rm x}\nolimits} _1}\left[ {\begin{array}{*{20}{c}}1\\{ - 3}\end{array}} \right] + {{\mathop{\rm x}\nolimits} _2}\left[ {\begin{array}{*{20}{c}}2\\{ - 8}\end{array}} \right] + {{\mathop{\rm x}\nolimits} _3}\left[ {\begin{array}{*{20}{c}}{ - 3}\\7\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}1\\1\end{array}} \right]\), convert it into the augmented matrix shown below:

\(\left[ {\begin{array}{*{20}{c}}1&2&{ - 3}&1\\{ - 3}&{ - 8}&7&1\end{array}} \right]\)

02

Apply the row operation

Perform an elementary row operation to produce the row-reduced echelon form of the matrix.

At row 2, multiply row 1 by 3 and add it to row 2.

\( \sim \left( {\begin{array}{*{20}{c}}1&2&{ - 3}&1\\0&{ - 2}&{ - 2}&4\end{array}} \right)\)

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

\( \sim \left( {\begin{array}{*{20}{c}}1&2&{ - 3}&1\\0&1&1&{ - 2}\end{array}} \right)\)

At row 1, multiply row 2 by 2 and subtract it from row 1.

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

Hence, you can consider \({{\mathop{\rm x}\nolimits} _1} = 5 + 5{{\mathop{\rm x}\nolimits} _3}\) and \({{\mathop{\rm x}\nolimits} _2} = - 2 - {{\mathop{\rm x}\nolimits} _3}\), where \({{\mathop{\rm x}\nolimits} _3}\) is any real number.

03

Determine two different ways to express \(\left[ {\begin{array}{*{20}{c}}1\\1\end{array}} \right]\) as a linear combination

Let \({{\mathop{\rm x}\nolimits} _3} = 0\) and \({{\mathop{\rm x}\nolimits} _3} = 1\) give two different ways to express \(\left( {\begin{array}{*{20}{c}}1\\1\end{array}} \right)\) as a linear combination of the vectors \(5{{\mathop{\rm v}\nolimits} _1} - 2{{\mathop{\rm v}\nolimits} _2}\) and \(10{{\mathop{\rm v}\nolimits} _1} - 3{{\mathop{\rm v}\nolimits} _2} + {{\mathop{\rm v}\nolimits} _3}\).

Thus, the two ways to express \(\left( {\begin{array}{*{20}{c}}1\\1\end{array}} \right)\) as a linear combination are \(5{{\mathop{\rm v}\nolimits} _1} - 2{{\mathop{\rm v}\nolimits} _2}\) and \(10{{\mathop{\rm v}\nolimits} _1} - 3{{\mathop{\rm v}\nolimits} _2} + {{\mathop{\rm v}\nolimits} _3}\).

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Most popular questions from this chapter

Consider the polynomials \({p_{\bf{1}}}\left( t \right) = {\bf{1}} + {t^{\bf{2}}},{p_{\bf{2}}}\left( t \right) = {\bf{1}} - {t^{\bf{2}}}\). Is \(\left\{ {{p_{\bf{1}}},{p_{\bf{2}}}} \right\}\) a linearly independent set in \({{\bf{P}}_{\bf{3}}}\)? Why or why not?

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.

A scientist solves a nonhomogeneous system of ten linear equations in twelve unknowns and finds that three of the unknowns are free variables. Can the scientist be certain that, if the right sides of the equations are changed, the new nonhomogeneous system will have a solution? Discuss.

Is it possible for a nonhomogeneous system of seven equations in six unknowns to have a unique solution for some right-hand side of constants? Is it possible for such a system to have a unique solution for every right-hand side? Explain.

Question: Determine if the matrix pairs in Exercises 19-22 are controllable.

21. (M) \(A = \left( {\begin{array}{*{20}{c}}0&1&0&0\\0&0&1&0\\0&0&0&1\\{ - 2}&{ - 4.2}&{ - 4.8}&{ - 3.6}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\0\\0\\{ - 1}\end{array}} \right)\).

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