Determine if the statement is true of false and justify your

Determine if the statement is true of false, and justify your answer. If U is a change of basis matrix between bases B_1 and B_2 of R^n, then U must be invertible. True. Since U = UW^-1 where V has columns given by the basis vectors of B_1, and w has columns given by the basis vectors of B_2, it follows that U is invertible, with U^-1 = WV^-1. False. Consider B_1 = {[1 1]} and B_2 {[1 0]}. True. Since U = W^-1V where V has columns given by the basis vectors of B_1, and W has columns given by the basis vectors of B_2, it follows that U is invertible, with U^-1 = V^-1W. False. Consider B_1 = {[1 1], [1 -1]} and B_2 = {[2 1], [1 1]}.

Solution

If u is a change of basis matrix between bases B1 and B2 of Rⁿ then u must be invertible:

True, since U = W^-1V where v has columns given by the basis vectors of B1 and W has columns given by the basis vectors of B2, it follows that U is invertible with U^-1 = v^-1 w