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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

In Exercises 19 and 20, \(V\) is a vector space. Mark each statement True or False. Justify each answer.

19.

a. The number of pivot columns of a matrix equals the dimension of its column space.

b. A plane in \({\mathbb{R}^3}\) is a two-dimensional subspace of \({\mathbb{R}^3}\).

c. The dimension of the vector space \({{\mathop{\rm P}\nolimits} _4}\) is 4.

d. If \(\dim V = n\) and \(S\) is a linearly independent set in \(V\), then \(S\) is a basis for \(V\).

e. If a set \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) spans a finite-dimensional vector space \(V\) and if \(T\) is a set of more than p vectors in \(V\), then \(T\) is linearly dependent.

If A is a \({\bf{7}} \times {\bf{5}}\) matrix, what is the largest possible rank of A? If Ais a \({\bf{5}} \times {\bf{7}}\) matrix, what is the largest possible rank of A? Explain your answer.

In Exercises 29 and 30, V is a nonzero finite-dimensional vector space, and the vectors listed belong to V. Mark each statement True or False. Justify each answer. (These questions are some difficult than those 19 and 20.)

30.

a. If there exists a linearly dependent set \(\left\{ {{{\mathop{\rm v}\nolimits} _1},...,{{\mathop{\rm v}\nolimits} _p}} \right\}\) in \(V\), then \(\dim V \le p\).

b. If every set of \(p\) elements in \(V\) fails to span \(V\), then \(\dim V > p\).

c. If \(p \ge 2\), and \(\dim V = p\), then every set of \(p - 1\) nonzero vectors is linearly independent.

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 \({\left( {\bf{u}} \right)_B} = y\).

Qis any matrix such that

\({\left( {\bf{v}} \right)_C} = Q{\left( {\bf{v}} \right)_B}\)for each v in V (9)

Set \({\bf{v}} = {{\bf{b}}_{\bf{1}}}\) in (9). Then (9) shows that \({\left( {{{\bf{b}}_{\bf{1}}}} \right)_C}\) is the first column of Q because (a) _____. Similarly, for \(k = {\bf{2}}\),鈥�..n the kth column of Q is (b) _____ because (c) _____. This shows the matrix \(\mathop P\limits_{C \leftarrow B} \) defined by (5) in Theorem 15 is the only matrix that satisfies condition (4).
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