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The vector-space [itex]\mathcal{F}([0,\pi],\mathbb{R})[/itex] consists of all real functions on [itex][0,\pi][/itex]. We let [itex]W[/itex] be its subspace with the basis [itex]\mathcal{B}[/itex] = {[itex]1,cost,cos(2t),cos(3t),...,cos(7t)[/itex]}.

[itex]T: W \rightarrow \mathbb{R} ^8[/itex] is the transformation where: [itex]T(h) = (h(t_1), h(t_2),...,h(t_8))[/itex], [itex]h \in W[/itex] and [itex]t_i = (\pi(2i-1))/16[/itex] , [itex]i\in[/itex]{[itex]1,2,...8[/itex]}

The call the standard basis for [itex]\mathbb{R} ^8[/itex] for [itex]\mathcal{C}[/itex]. The change-of-coordinates matrix of [itex]T[/itex] from [itex]\mathcal{B}[/itex] to [itex]\mathcal{C}[/itex] is an invertibel matrix we call [itex]M[/itex].

My question is how I can show that [itex]T[/itex] is an isomorphism. I know that this means that [itex]T[/itex] must be a invertibel linear transformation, but how do I show this? Does it have anything to do with the fact that the change-of-coordinates matrix [itex]M[/itex] is invertibel, and how?

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# Homework Help: Linear algebra, isomorphism

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