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Chapter 2: Q2.9-29E (page 93)

[M] Let \(H = {\bf{span}}\left\{ {{{\bf{v}}_1},{{\bf{v}}_2}} \right\}\) and \({\rm B} = \left\{ {{{\bf{v}}_{\bf{1}}},{{\bf{v}}_{\bf{2}}}} \right\}\). Show that \(x\) is in H, and find the \(\beta - \)coordinate vector of x, when

\({{\bf{v}}_{\bf{1}}} = \left[ {\begin{array}{*{20}{c}}{{\bf{11}}}\\{ - {\bf{5}}}\\{{\bf{10}}}\\{\bf{7}}\end{array}} \right]\), \({{\bf{v}}_{\bf{2}}} = \left[ {\begin{array}{*{20}{c}}{{\bf{14}}}\\{ - {\bf{8}}}\\{{\bf{13}}}\\{{\bf{10}}}\end{array}} \right]\), \({\bf{x}} = \left[ {\begin{array}{*{20}{c}}{{\bf{19}}}\\{ - {\bf{13}}}\\{{\bf{18}}}\\{{\bf{15}}}\end{array}} \right]\)

Short Answer

Expert verified

The \(\beta - \)coordinates of \({\bf{x}}\) are \(\left( { - \frac{5}{3},\frac{8}{3}} \right)\).

Step by step solution

01

Form a matrix using the vectors

The matrix formed using thecolumn vectors is

\(A = \left[ {\begin{array}{*{20}{c}}{11}&{14}&{19}\\{ - 5}&{ - 8}&{ - 13}\\{10}&{13}&{18}\\7&{10}&{15}\end{array}} \right]\).

02

Convert the matrix into the row-reduced echelon form

Consider matrix \(A = \left[ {\begin{array}{*{20}{c}}{11}&{14}&{19}\\{ - 5}&{ - 8}&{ - 13}\\{10}&{13}&{18}\\7&{10}&{15}\end{array}} \right]\).

Use the code in MATLAB to obtain therow-reducedechelon form as shown below:

\[\begin{array}{l} > > {\rm{ A }} = {\rm{ }}\left[ {{\rm{ }}\begin{array}{*{20}{c}}{11}&{14}&{19;\,\,\begin{array}{*{20}{c}}{ - 5}&{ - 8}&{ - 13;\,\,\begin{array}{*{20}{c}}{10}&{13}&{18;\;\begin{array}{*{20}{c}}7&{10}&{15}\end{array}}\end{array}}\end{array}}\end{array}} \right];\\ > > {\rm{ U}} = {\rm{rref}}\left( {\rm{A}} \right)\end{array}\]

\(\left[ {\begin{array}{*{20}{c}}{11}&{14}&{19}\\{ - 5}&{ - 8}&{ - 13}\\{10}&{13}&{18}\\7&{10}&{15}\end{array}} \right] \sim \left[{\begin{array}{*{20}{c}}1&0&{ - \frac{5}{3}}\\0&1&{\frac{8}{3}}\\0&0&0\\0&0&0\end{array}} \right]\)

03

Find the coordinates

The equation \({c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} = {\bf{x}}\) is consistent.The values of \({c_1}\) and \({c_2}\) from the echelon form are

\({c_1} = - \frac{5}{3}\)and \({c_2} = \frac{8}{3}\).

So, the \(\beta - \)coordinates of \({\bf{x}}\) are \(\left( { - \frac{5}{3},\frac{8}{3}} \right)\).

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Most popular questions from this chapter

Use matrix algebra to show that if A is invertible and D satisfies \(AD = I\) then \(D = {A^{ - {\bf{1}}}}\).

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 \(E = \left( {\begin{aligned}{*{20}{c}}{ - 5}\\3\end{aligned}} \right)\)

\(A + 2B\), \(3C - E\), \(CB\), \(EB\).

Give a formula for \({\left( {ABx} \right)^T}\), where \({\bf{x}}\) is a vector and \(A\) and \(B\) are matrices of appropriate sizes.

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.

37. Construct a random \({\bf{4}} \times {\bf{4}}\) matrix Aand test whether \(\left( {A + I} \right)\left( {A - I} \right) = {A^2} - I\). The best way to do this is to compute \(\left( {A + I} \right)\left( {A - I} \right) - \left( {{A^2} - I} \right)\) and verify that this difference is the zero matrix. Do this for three random matrices. Then test \(\left( {A + B} \right)\left( {A - B} \right) = {A^2} - {B^{\bf{2}}}\) the same way for three pairs of random \({\bf{4}} \times {\bf{4}}\) matrices. Report your conclusions.

Suppose the third column of Bis the sum of the first two columns. What can you say about the third column of AB? Why?
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