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I'm working on an example question with the following info:

[itex]\alpha[/itex] = {(3,0,1) , (3,1,1), (2,1,1)} [itex]\beta[/itex] = {(1,1), (1,-1)} Are a set of bases. [T]_{[itex]\beta\alpha[/itex]}= \begin{bmatrix} 1 & 2 & -1\\ 0 & 1 & -1 \end{bmatrix} Now they go on to say:

Let T: R^{3}--> R^{3}be the transformation whose matris with respect to the basis [itex]\alpha[/itex] is:

[T]_{[itex]\\alpha\alpha[/itex]}= \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}

Now I'm trying to do some back calculations to figure out how they got that matrix, but the only thing I know would be to use a calculation such as_{[itex]\beta\alpha[/itex] }[T]_{[itex]\alpha\beta[/itex]}

But I can't write out a vector in R^{2}as a linear combination of vectors in R^{3}right? So how would I get that

[T]_{[itex]\\alpha\alpha[/itex]}= \begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{bmatrix}

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# Change of Basis between different size spaces

Can you offer guidance or do you also need help?

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