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Chapter 4: Q15E (page 191)

Exercises 15 and 16 provide a proof of Theorem 15. Fill in a justification for each step. 15.

Given v inV , there exist scalars , such that because . Apply the coordinate mapping determined by the basis , and obtain because \({\rm{v}}\)\((b)\underline {\,\,\,\,\,\,\,\,\,\,\,\,} \) . This equation may be written in the form \({\left( v \right)_C} = \left( {{{\left( {{b_1}} \right)}_C}\,\,\,{{\left( {{b_2}} \right)}_C}\,\,\,...\,\,\,\,{{\left( {{b_n}} \right)}_C}} \right)\left( \begin{array}{l}{x_1}\\\,\, \vdots \\{x_n}\end{array} \right)\)by the definition of \((c)\underline {\,\,\,\,\,\,\,\,\,\,\,\,} \) . This shows that the matrix \(\mathop P\limits_{C \to B} \)shown in (5) satisfies \({\left( v \right)_C} = \mathop P\limits_{C \to B} {\left( v \right)_B}\)each \(v\) in \(V\) , because the vector on the right side of (8) is \((d)\underline {\,\,\,\,\,\,\,\,\,\,\,\,} \).

Short Answer

Expert verified

(a) \(B\) is the basis for vector space \(V\)

(b) coordinate mapping is a linear transformation

(c) product of matrix and vector

(d) the coordinate vector of \(v\) relative to \(B\)

Step by step solution

01

State the theorem for change of coordinate matrix from \(B\) to \(C\) 

If \(B = \left\{ {{b_1}.....{b_n}} \right\}\) and \(C = \left\{ {{c_1}.....{c_n}} \right\}\) are the bases for vector space \(V\), then according to the theorem of change of coordinates matrix from \(B\) to \(C\), there is always a matrix \(\mathop P\limits_{C \to B} \) of dimension \(n \times n\) such that \({\left( x \right)_c} = \mathop P\limits_{C \to B} {\left( x \right)_B}\). It means that the vectors in basis \(B\) have \(C\) coordinate vectors, which are the same as the columns of \(\mathop P\limits_{C \to B} \).

02

Use the first line of the statement to fill the gap in (a).

As \(B = \left\{ {{b_1}.....{b_n}} \right\}\) and \(C = \left\{ {{c_1}.....{c_n}} \right\}\) are the bases for vector space \(V\), we can write \(v = {x_1}{b_1} + {x_2}{b_2} + .... + {x_n}{b_n}\).Thus, 鈥淺(B\) is the basis for vector space \(V\)鈥� fills in blank (a).

03

Use the last line of the statement to fill the gap in (b).

As the vectors in basis \(B\) have \(C\) coordinate vectors, which are the same as the columns of\(\mathop P\limits_{C \to B} \), coordinate mapping must be a linear transformation. Thus, 鈥渃oordinate mapping is a linear transformation鈥� fills in blank (b).

04

Use the definition of the product of matrix and vector

The equation of the form \({\left( v \right)_C} = \left( {{{\left( {{b_1}} \right)}_C}\,\,\,{{\left( {{b_2}} \right)}_C}\,\,\,...\,\,\,\,{{\left( {{b_n}} \right)}_C}} \right)\left( \begin{array}{l}{x_1}\\\,\, \vdots \\{x_n}\end{array} \right)\) is a product of a matrix and a vector. The matrix is \(\left( {{{\left( {{b_1}} \right)}_C}\,\,\,{{\left( {{b_2}} \right)}_C}\,\,\,...\,\,\,\,{{\left( {{b_n}} \right)}_C}} \right)\) , and the vector is \(\left( \begin{array}{l}{x_1}\\\,\, \vdots \\{x_n}\end{array} \right)\).

Thus, 鈥減roduct of matrix and vector鈥� fills in blank (c).

05

Use the last line of the statement to fill the gap in (d)

There is always a matrix \(\mathop P\limits_{C \to B} \) of the dimension \(n \times n\) such that \({\left( x \right)_c} = \mathop P\limits_{C \to B} {\left( x \right)_B}\). It implies that the coordinate vector of x relative to \(B\) is changed to \(C\).

So, 鈥渢he coordinate vector of \(v\) relative to \(B\)鈥� fills in blank (d).

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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?

Let be a basis of\({\mathbb{R}^n}\). .Produce a description of an \(B = \left\{ {{{\bf{b}}_{\bf{1}}},....,{{\bf{b}}_n}\,} \right\}\)matrix A that implements the coordinate mapping \({\bf{x}} \mapsto {\left( {\bf{x}} \right)_B}\). Find it. (Hint: Multiplication by A should transform a vector x into its coordinate vector \({\left( {\bf{x}} \right)_B}\)). (See Exercise 21.)

Let \({\mathop{\rm u}\nolimits} = \left[ {\begin{array}{*{20}{c}}1\\2\end{array}} \right]\). Find \({\mathop{\rm v}\nolimits} \) in \({\mathbb{R}^3}\) such that \(\left[ {\begin{array}{*{20}{c}}1&{ - 3}&4\\2&{ - 6}&8\end{array}} \right] = {{\mathop{\rm uv}\nolimits} ^T}\) .

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

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

Which of the subspaces \({\rm{Row }}A\) , \({\rm{Col }}A\), \({\rm{Nul }}A\), \({\rm{Row}}\,{A^T}\) , \({\rm{Col}}\,{A^T}\) , and \({\rm{Nul}}\,{A^T}\) are in \({\mathbb{R}^m}\) and which are in \({\mathbb{R}^n}\) ? How many distinct subspaces are in this list?.
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