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

Given vectors, \({u_{\bf{1}}}\),….,\({u_p}\) and w in V, show that w is a linear combination of \({u_{\bf{1}}}\),….,\({u_p}\) if and only if \({\left( w \right)_B}\) is a linear combination of vectors \({\left( {{{\bf{u}}_{\bf{1}}}} \right)_B}\),….,\({\left( {{{\bf{u}}_p}} \right)_B}\).

Short Answer

Expert verified

\({\left( {\bf{w}} \right)_B}\) can be written as a linear combination of \({\left( {{{\bf{u}}_1}} \right)_B}\), \({\left( {{{\bf{u}}_2}} \right)_B}\),….,\({\left( {{{\bf{u}}_p}} \right)_B}\).

Step by step solution

01

Write w as a linear combination

Was a linear combinationof \({{\bf{u}}_1}\), \({{\bf{u}}_2}\),…., \({{\bf{u}}_p}\) can be written as:

\({\bf{w}} = {c_1}{{\bf{u}}_1} + {c_2}{{\bf{u}}_2} + .... + {c_p}{{\bf{u}}_p}\)

The first equation has a trivial solution if the second equation also has a trivial solution.

02

Check for \({\left( {\bf{w}} \right)_B}\)

If B is a basis for space V, then the coordinate mapping is one to one transformation. Therefore,

\(\begin{array}{c}{\left( {\bf{w}} \right)_B} = {\left( {{c_1}{{\bf{u}}_1} + {c_2}{{\bf{u}}_2} + .....{c_p}{{\bf{u}}_p}} \right)_B}\\ = {\left( {{c_1}{{\bf{u}}_1}} \right)_B} + {\left( {{c_2}{{\bf{u}}_2}} \right)_B} + .... + {\left( {{c_p}{{\bf{u}}_p}} \right)_B}\\ = {c_1}{\left( {{{\bf{u}}_1}} \right)_B} + {c_2}{\left( {{{\bf{u}}_2}} \right)_B} + .... + {c_p}{\left( {{{\bf{u}}_p}} \right)_B}\end{array}\)

Thus, the \({\left( {\bf{w}} \right)_B}\) can be written as a linear combination of \({\left( {{{\bf{u}}_1}} \right)_B}\), \({\left( {{{\bf{u}}_2}} \right)_B}\),….,\({\left( {{{\bf{u}}_p}} \right)_B}\).

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Most popular questions from this chapter

In Exercise 1, find the vector x determined by the given coordinate vector \({\left( x \right)_{\rm B}}\) and the given basis \({\rm B}\).

1. \({\rm B} = \left\{ {\left( {\begin{array}{*{20}{c}}{\bf{3}}\\{ - {\bf{5}}}\end{array}} \right),\left( {\begin{array}{*{20}{c}}{ - {\bf{4}}}\\{\bf{6}}\end{array}} \right)} \right\},{\left( x \right)_{\rm B}} = \left( {\begin{array}{*{20}{c}}{\bf{5}}\\{\bf{3}}\end{array}} \right)\)

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.

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

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

If the null space of a\({\bf{5}} \times {\bf{6}}\)matrix A is 4-dimensional, what is the dimension of the row space of A?

In Exercise 7, find the coordinate vector \({\left( x \right)_{\rm B}}\) of x relative to the given basis \({\rm B} = \left\{ {{b_{\bf{1}}},...,{b_n}} \right\}\).

7. \({b_{\bf{1}}} = \left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{1}}}\\{ - {\bf{3}}}\end{array}} \right),{b_{\bf{2}}} = \left( {\begin{array}{*{20}{c}}{ - {\bf{3}}}\\{\bf{4}}\\{\bf{9}}\end{array}} \right),{b_{\bf{3}}} = \left( {\begin{array}{*{20}{c}}{\bf{2}}\\{ - {\bf{2}}}\\{\bf{4}}\end{array}} \right),x = \left( {\begin{array}{*{20}{c}}{\bf{8}}\\{ - {\bf{9}}}\\{\bf{6}}\end{array}} \right)\)

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