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Chapter 4: Q14E (page 191)

In \({{\rm{P}}_2}\), find the change-of-coordinates matrix from the basis \(B = \left\{ {1 - 3t,\,2 + t - 5{t^2},\,1 + 2t} \right\}\) to the standard basis. Then write \({t^2}\) as a linear combination of \(B\).

Short Answer

Expert verified

\(\left( {\begin{array}{*{20}{c}}1&0&0&3\\0&1&0&{ - 2}\\0&0&0&1\end{array}} \right)\), \({t^2} = 3\left( {1 - 3{t^2}} \right) - 2\left( {2 + t - 5{t^2}} \right) + \left( {1 + 2t} \right)\)

Step by step solution

01

Write the \(C\)-coordinate vector of \({b_1}\), \({b_2}\), \({b_3}\) and then write  the \(\mathop P\limits_{C \to B} \) matrix

The \(C\)-coordinate vectors of \({b_1}\),\({b_2}\),\({b_3}\) are\({\left( {{b_1}} \right)_c} = \left( \begin{array}{l}\,\,\,\,\,\,1\,\\\,\,\,\,\,\,0\\\,\,\, - 3\end{array} \right)\), \({\left( {{b_2}} \right)_c} = \left( \begin{array}{l}\,\,\,2\\\,\,\,1\\ - 5\end{array} \right)\), \({\left( {{b_3}} \right)_c} = \left( \begin{array}{l}\,\,\,1\,\\\,\,\,2\\\,\,\,0\end{array} \right)\). Thus, the \(\mathop P\limits_{C \to B} \) matrix can be written as \(\mathop P\limits_{C \to B} = \left( {\begin{array}{*{20}{c}}1&2&1\\0&1&2\\3&{ - 5}&0\end{array}} \right)\).

02

Use the result \({\left( x \right)_C} = P{\left( x \right)_B}\)

If \(x\)is the vector \(\left\{ {{t^2}} \right\}\), then according to the result \({\left( x \right)_c} = \mathop P\limits_{C \to B} {\left( x \right)_B}\), you can write

\(\mathop P\limits_{C \to B} {\left( x \right)_B} = \left( \begin{array}{l}0\\0\\1\end{array} \right)\).

03

Substitute the matrix for\(\mathop P\limits_{C \to B} \)and apply row reduction on the augmented matrix

\(\left( {\begin{array}{*{20}{c}}1&2&1&0\\0&1&2&0\\{ - 3}&{ - 5}&0&1\end{array}} \right) \sim \left( {\begin{array}{*{20}{c}}1&0&0&3\\0&1&0&{ - 2}\\0&0&0&1\end{array}} \right)\)

04

Compare the resulting matrix with \(P{\left( x \right)_B}\) and write the column matrix\({\left( x \right)_B}\)

\({\left( x \right)_B} = \left( \begin{array}{l}\,\,\,3\\ - 2\\\,\,\,1\end{array} \right)\)

05

Write \({t^2}\) as a linear combination of \(B\) using the \({\left( x \right)_B}\)matrix

\({t^2} = 3\left( {1 - 3{t^2}} \right) - 2\left( {2 + t - 5{t^2}} \right) + \left( {1 + 2t} \right)\)

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Most popular questions from this chapter

Exercises 23-26 concern a vector space V, a basis \(B = \left\{ {{{\bf{b}}_{\bf{1}}},....,{{\bf{b}}_n}\,} \right\}\) and the coordinate mapping \({\bf{x}} \mapsto {\left( {\bf{x}} \right)_B}\).

Show the coordinate mapping is one to one. (Hint: Suppose \({\left( {\bf{u}} \right)_B} = {\left( {\bf{w}} \right)_B}\) for some u and w in V, and show that \({\bf{u}} = {\bf{w}}\)).

Let S be a maximal linearly independent subset of a vector space V. In other words, S has the property that if a vector not in S is adjoined to S, the new set will no longer be linearly independent. Prove that S must be a basis of V. [Hint: What if S were linearly independent but not a basis of V?]

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

20. \(A = \left( {\begin{array}{*{20}{c}}{.8}&{ - .3}&0\\{.2}&{.5}&1\\0&0&{ - .5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\1\\0\end{array}} \right)\).

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

14. Show that if \(Q\) is an invertible, then \({\mathop{\rm rank}\nolimits} AQ = {\mathop{\rm rank}\nolimits} A\). (Hint: Use Exercise 13 to study \({\mathop{\rm rank}\nolimits} {\left( {AQ} \right)^T}\).)
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