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Chapter 4: Q18Q (page 191)

Let \(B = \left\{ {{{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _n}} \right\}\) be a basis for a vector space \(V\). Explain why the \(B - \)coordinate vectors of \({{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _n}\) are the columns \({{\mathop{\rm e}\nolimits} _1},...,{{\mathop{\rm e}\nolimits} _n}\) of the \(n \times n\) identity matrix.

Answer:

The \(B - \)coordinate vectors of \({{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _n}\) are columns \({{\mathop{\rm e}\nolimits} _1},...,{{\mathop{\rm e}\nolimits} _n}\) of the \(n \times n\) identity matrix.

Short Answer

Expert verified

The \(B - \)coordinate vectors of \({{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _n}\) are columns \({{\mathop{\rm e}\nolimits} _1},...,{{\mathop{\rm e}\nolimits} _n}\) of the \(n \times n\) identity matrix.

Step by step solution

01

State the B-coordinate vector

Suppose \(B = \left\{ {{{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _n}} \right\}\) is a basis for \(V\) and x is in \(V\). Thecoordinates of \({\mathop{\rm x}\nolimits} \) relative tobasis \(B\)(or the \(B\)-coordinates of x) are the weights \({c_1},...,{c_n}\), such that \({\mathop{\rm x}\nolimits} = {c_1}{b_1} + ... + {c_n}{b_n}\).

02

Explain that the \(B - \)coordinate vectors of \({{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _n}\) are columns \({{\mathop{\rm e}\nolimits} _1},...,{{\mathop{\rm e}\nolimits} _n}\) of the identity matrix 

For each \(k\), \({{\mathop{\rm b}\nolimits} _k} = 0 \cdot {{\mathop{\rm b}\nolimits} _1} + \cdots + 1 \cdot {{\mathop{\rm b}\nolimits} _k} + \cdots + 0 \cdot {{\mathop{\rm b}\nolimits} _n}\).

Therefore, \({\left( {{{\mathop{\rm b}\nolimits} _k}} \right)_B} = \left( {0,...,1,...,0} \right) = {{\mathop{\rm e}\nolimits} _k}\).

Thus, the \(B - \)coordinate vectors of \({{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _n}\) are columns \({{\mathop{\rm e}\nolimits} _1},...,{{\mathop{\rm e}\nolimits} _n}\) of the \(n \times n\) identity matrix.

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Most popular questions from this chapter

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?.

Define by \(T\left( {\mathop{\rm p}\nolimits} \right) = \left( {\begin{array}{*{20}{c}}{{\mathop{\rm p}\nolimits} \left( 0 \right)}\\{{\mathop{\rm p}\nolimits} \left( 1 \right)}\end{array}} \right)\). For instance, if \({\mathop{\rm p}\nolimits} \left( t \right) = 3 + 5t + 7{t^2}\), then \(T\left( {\mathop{\rm p}\nolimits} \right) = \left( {\begin{array}{*{20}{c}}3\\{15}\end{array}} \right)\).

  1. Show that \(T\) is a linear transformation. (Hint: For arbitrary polynomials p, q in \({{\mathop{\rm P}\nolimits} _2}\), compute \(T\left( {{\mathop{\rm p}\nolimits} + {\mathop{\rm q}\nolimits} } \right)\) and \(T\left( {c{\mathop{\rm p}\nolimits} } \right)\).)
  2. Find a polynomial p in \({{\mathop{\rm P}\nolimits} _2}\) that spans the kernel of \(T\), and describe the range of \(T\).

Let \(B = \left\{ {\left( {\begin{array}{*{20}{c}}{\bf{1}}\\{ - {\bf{4}}}\end{array}} \right),\,\left( {\begin{array}{*{20}{c}}{ - {\bf{2}}}\\{\bf{9}}\end{array}} \right)\,} \right\}\). Since the coordinate mapping determined by B is a linear transformation from \({\mathbb{R}^{\bf{2}}}\) into \({\mathbb{R}^{\bf{2}}}\), this mapping must be implemented by some \({\bf{2}} \times {\bf{2}}\) matrix A. Find it. (Hint: Multiplication by A should transform a vector x into its coordinate vector \({\left( {\bf{x}} \right)_B}\)).

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 the null space of an \({\bf{8}} \times {\bf{5}}\) matrix A is 2-dimensional, what is the dimension of the row space of A?
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