Question

In Exercises 11 and 12, and Care bases for a vector space V. Mark each statement True or False. Justify each answer.\(B\)

a. The columns of the change-of-coordinates matrix\(\mathop P\limits_{C \to B} \)are\(B\)-coordinate vectors of the vectors in\(C\).

b. If \(V = {R^n}\)and \(C\) is the standard basis for \(V\) , then \(\mathop P\limits_{C \to B} \)is the same as the change-of-coordinates matrix \({P_B}\) introduced in section 4.4.

Short answer

(a) False

The columns of the change-of-coordinates matrix \(\mathop P\limits_{C \to B} \)are the \(C\)-coordinate vectors of the vectors in \(B\).

(b) True

As \({\left( {{b_1}} \right)_\xi } = {b_1}\), \(\mathop P\limits_{\xi \to B} \) is the same as the change of coordinates matrix \({P_B}\) defines as \({P_B} = \left( {{b_1},{b_2}......{b_n}} \right)\).

Step-by-step solution

Assume \(B\) and \(C\) to be the bases for vector space \(V\)and \(\xi \) to be a standard basis in \({\mathbb{R}^n}\)

Suppose \(B = \left\{ {{b_1}.....{b_n}} \right\}\), \(C = \left\{ {{c_1}.....{c_n}} \right\}\) are the bases for vector space \(V\) and \(\xi = \left\{ {{e_1}......{e_n}} \right\}\) is the standard basis in \({\mathbb{R}^n}\).

Use the theorem for change of coordinates matrix from \(B\) to \(C\) and the concept of change of basis in \({\mathbb{R}^n}\)

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 the 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 vector, which are the same as the columns of \(\mathop P\limits_{C \to B} \).

If \(B = \left\{ {{b_1}.....{b_n}} \right\}\) and \(\xi = \left\{ {{e_1}......{e_n}} \right\}\) are the standard bases in \({\mathbb{R}^n}\), then \({\left( {{b_1}} \right)_\xi } = {b_1}\). So, \(\mathop P\limits_{\xi \to B} \) is the same as the change of coordinates matrix \({P_B}\) defines as \({P_B} = \left( {{b_1},{b_2}......{b_n}} \right)\).

Draw a conclusion

So, statement (a) is false and statement (b) is true.