Question

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

a. The columns of \(\mathop P\limits_{C \to B} \) are linearly independent.

b. If \(V = {\mathbb{R}^2}\), \(B = \left\{ {{b_1},{b_2}} \right\}\),\(C = \left\{ {{c_1},{c_2}} \right\}\) then row reduction of \(\left( {{c_1}\,\,\,\,{c_2}\,\,\,{b_1}\,\,\,{b_2}} \right)\)to \(\left( {I\,\,\,\,\,P} \right)\)produces a matrix \(P\) that satisfies \({\left( x \right)_B} = P{\left( x \right)_C}\) for all \(x\) in \(V\).

Short answer

(a) True

The columns of matrix \(\mathop P\limits_{C \to B} \) are always linearly independent, which are also the vectors in basis \(B\).

(b) False

The reduction of\(\left( {{c_1}\,\,\,\,{c_2}\,\,\,{b_1}\,\,\,{b_2}} \right)\)to\(\left( {I\,\,\,\,\,P} \right)\)produces matrix\(P\)that satisfies\({\left( x \right)_C} = P{\left( x \right)_B}\)for all\(x\)in\(V\).

Step-by-step solution

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

Assume \(B = \left\{ {{b_1}.....{b_n}} \right\}\) and \(C = \left\{ {{c_1}.....{c_n}} \right\}\) 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 row reduction

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 \(C\), the vectors in basis \(B\) have \(C\) coordinate vector, which are the same as the columns of \(\mathop P\limits_{C \to B} \). Thus, these columns are linearly independent.

If \(B = \left\{ {{b_1}.....{b_n}} \right\}\) and \(C = \left\{ {{c_1}.....{c_n}} \right\}\) are the bases in \(V = {\mathbb{R}^2}\), then according to the concept of row reduction, the reduction of\(\left( {{c_1}\,\,\,\,{c_2}\,\,\,{b_1}\,\,\,{b_2}} \right)\)to\(\left( {I\,\,\,\,\,P} \right)\)produces matrix\(P\)that satisfies\({\left( x \right)_C} = P{\left( x \right)_B}\)for all\(x\)in\(V\).

Draw a conclusion

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