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Chapter 4: Q24E (page 191)

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 that the coordinate mapping is onto \({\mathbb{R}^n}\). That is, given any y in \({\mathbb{R}^n}\), with entries \({y_{\bf{1}}}\),鈥�.,\({y_n}\), produce u in V such that \({\left( {\bf{u}} \right)_B} = y\).

Short Answer

Expert verified

The given mapping is onto.

Step by step solution

01

Find the coordinate u

\(y\)in \({\mathbb{R}^n}\) is expressed as:

\(y = \left( {\begin{array}{*{20}{c}}{{y_1}}\\{{y_2}}\\{{y_3}}\end{array}} \right)\)

The coordinates of \({\bf{u}}\) relative to basis B are the weights \({y_1}\), \({y_2}\),鈥� \({y_n}\) such that

\({\bf{u}} = {y_1}{{\bf{b}}_1} + {y_2}{{\bf{b}}_2} + .... + {y_n}{{\bf{b}}_n}\)

02

Find the vector \({\left( {\bf{u}} \right)_B}\)

The coordinate of vector u is:

\(\begin{array}{c}{\left( {\bf{u}} \right)_B} = \left( {\begin{array}{*{20}{c}}{{y_1}}\\{{y_2}}\\{{y_3}}\end{array}} \right)\\ = y\end{array}\)

For any arbitrary element \(y \in {\mathbb{R}^n}\), there is an element u in V. So, every vector in \({\mathbb{R}^n}\) has a pre-image corresponding to the mapping.

Thus, the coordinate mapping is onto.

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Most popular questions from this chapter

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

16. If \(A\) is an \(m \times n\) matrix of rank\(r\), then a rank factorization of \(A\) is an equation of the form \(A = CR\), where \(C\) is an \(m \times r\) matrix of rank\(r\) and \(R\) is an \(r \times n\) matrix of rank \(r\). Such a factorization always exists (Exercise 38 in Section 4.6). Given any two \(m \times n\) matrices \(A\) and \(B\), use rank factorizations of \(A\) and \(B\) to prove that rank\(\left( {A + B} \right) \le {\mathop{\rm rank}\nolimits} A + {\mathop{\rm rank}\nolimits} B\).

(Hint: Write \(A + B\) as the product of two partitioned matrices.)

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

22. (M) \(A = \left( {\begin{array}{*{20}{c}}0&1&0&0\\0&0&1&0\\0&0&0&1\\{ - 1}&{ - 13}&{ - 12.2}&{ - 1.5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}1\\0\\0\\{ - 1}\end{array}} \right)\).

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 \({\bf{u}} = {\bf{w}}\)).

If A is a \({\bf{6}} \times {\bf{8}}\) matrix, what is the smallest possible dimension of Null A?

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.)
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