Question

Question: In Exercises 5 and 6, let \({{\bf{b}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{\bf{1}}\\{\bf{1}}\end{array}} \right)\), \({{\bf{b}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{\bf{0}}\\{ - {\bf{2}}}\end{array}} \right)\), and \({{\bf{b}}_{\bf{3}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{ - {\bf{5}}}\\{\bf{1}}\end{array}} \right)\) and \(S = \left\{ {{{\bf{b}}_{\bf{1}}},\,{{\bf{b}}_{\bf{2}}},\,{{\bf{b}}_{\bf{3}}}} \right\}\). Note that S is an orthogonal basis of \({\mathbb{R}^{\bf{3}}}\). Write each is given points as an affine combination of the points in the set S, if possible. (Hint: Use Theorem 5 in section 6.2 instead of row reduction to find the weights.)

a. \({{\bf{p}}_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{0}}\\{ - {\bf{19}}}\\{\bf{5}}\end{array}} \right)\) b. \({{\bf{p}}_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{{\bf{1}}.{\bf{5}}}\\{ - {\bf{1}}.{\bf{3}}}\\{ - .{\bf{5}}}\end{array}} \right)\) c. \({{\bf{p}}_{\bf{3}}} = \left( {\begin{array}{*{20}{c}}{\bf{5}}\\{ - {\bf{4}}}\\{\bf{0}}\end{array}} \right)\)

Short answer

a. \({{\bf{p}}_1} = - 4\left( {{{\bf{b}}_1}} \right) + 2\left( {{{\bf{b}}_2}} \right) + 3\left( {{{\bf{b}}_3}} \right) \in {\rm{aff}}\,\,S\), the sum of coefficients of in S is 1.

b. \({{\bf{p}}_2} = 0.2\left( {{{\bf{b}}_1}} \right) + 0.5\left( {{{\bf{b}}_2}} \right) + 0.3\left( {{{\bf{b}}_3}} \right) \in {\rm{aff}}\,S\), the sum of coefficients of in S is 1.

c. \({{\bf{p}}_3} = 1\left( {{{\bf{b}}_1}} \right) + 1\left( {{{\bf{b}}_2}} \right) + 1\left( {{{\bf{b}}_3}} \right)\), the sum of coefficients of in S is not 1.

Step-by-step solution

Find the augmented matrix

Write the augmented matrix by using the given points as shown below:

\(\begin{array}{c}M = \left( {\begin{array}{*{20}{c}}{{{\bf{b}}_1}}&{{{\bf{b}}_2}}&{{{\bf{b}}_3}}&{{{\bf{p}}_1}}&{{{\bf{p}}_2}}&{{{\bf{p}}_3}}\end{array}} \right)\\ = \left( {\begin{array}{*{20}{c}}2&1&2&0&{1.5}&5\\1&0&{ - 5}&{ - 19}&{ - 1.3}&{ - 4}\\1&{ - 2}&1&{ - 5}&{ - 0.5}&0\end{array}} \right)\end{array}\)

Write the row reduced form of the augmented matrix

Row reduce the augmented matrix as shown below:

\(\begin{array}{c}M = \left( {\begin{array}{*{20}{c}}0&1&{12}&{38}&{4.1}&{13}\\1&0&{ - 5}&{ - 19}&{ - 1.3}&{ - 4}\\0&{ - 2}&6&{14}&{0.8}&4\end{array}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ \begin{array}{l}{R_1} \to {R_1} - 2{R_2}\\{R_3} \to {R_3} - {R_2}\end{array} \right\}\\ = \left( {\begin{array}{*{20}{c}}1&0&{ - 5}&{ - 19}&{ - 1.3}&{ - 4}\\0&1&{12}&{38}&{4.1}&{13}\\0&{ - 2}&6&{14}&{0.8}&4\end{array}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {{R_1} \leftrightarrow {R_2}} \right\}\\ = \left( {\begin{array}{*{20}{c}}1&0&{ - 5}&{ - 19}&{ - 1.3}&{ - 4}\\0&1&{12}&{38}&{4.1}&{13}\\0&0&{30}&{90}&{9.0}&{30}\end{array}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {{R_3} \to {R_3} + 2{R_2}} \right\}\\ = \left( {\begin{array}{*{20}{c}}1&0&{ - 5}&{ - 19}&{ - 1.3}&{ - 4}\\0&1&{12}&{38}&{4.1}&{13}\\0&0&1&3&{0.3}&1\end{array}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ {{R_3} \to \frac{1}{{30}}{R_3}} \right\}\\ = \left( {\begin{array}{*{20}{c}}1&0&0&{ - 4}&{0.2}&1\\0&1&0&2&{0.5}&1\\0&0&1&3&{0.3}&1\end{array}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left\{ \begin{array}{l}{R_1} \to {R_1} + 5{R_3}\\{R_2} \to {R_2} - 12{R_3}\end{array} \right\}\end{array}\)

Check for the affine combination of \({{\bf{p}}_{\bf{1}}}\)

Use the augmented matrix, \({{\bf{p}}_1}\) which can be expressed as shown below:

\({{\bf{p}}_1} = - 4\left( {{{\bf{b}}_1}} \right) + 2\left( {{{\bf{b}}_2}} \right) + 3\left( {{{\bf{b}}_3}} \right)\)

The sum of the coefficients is \( - 4 + 2 + 3 = 1\).

So, \({{\bf{p}}_1}\) is an affine combination of point in S.

\({{\bf{p}}_1} = - 4\left( {{{\bf{b}}_1}} \right) + 2\left( {{{\bf{b}}_2}} \right) + 3\left( {{{\bf{b}}_3}} \right)\)

Check for the affine combination of \({{\bf{p}}_{\bf{2}}}\)

Use the augmented matrix, \({{\bf{p}}_2}\) which can be expressed as shown below:

\({{\bf{p}}_2} = 0.2\left( {{{\bf{b}}_1}} \right) + 0.5\left( {{{\bf{b}}_2}} \right) + 0.3\left( {{{\bf{b}}_3}} \right)\)

The sum of coefficients is \(0.2 + 0.5 + 0.3 = 1\).

So, \({{\bf{p}}_2}\) is an affine combination of point in S.

\({{\bf{p}}_2} = 0.2\left( {{{\bf{b}}_1}} \right) + 0.5\left( {{{\bf{b}}_2}} \right) + 0.3\left( {{{\bf{b}}_3}} \right)\)

Check for the affine combination of \({{\bf{p}}_{\bf{3}}}\)

Use the augmented matrix, \({{\bf{p}}_3}\) which can be expressed as shown below:

\({{\bf{p}}_3} = 1\left( {{{\bf{b}}_1}} \right) + 1\left( {{{\bf{b}}_2}} \right) + 1\left( {{{\bf{b}}_3}} \right)\)

The sum of coefficients is \(1 + 1 + 1 = 3 \ne 1\).

So, \({{\bf{p}}_3}\) is not an affine combination of point in S.