Change of Basis between different size spaces

[trap101]

Hi,

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: R³--> R³ 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 R² as a linear combination of vectors in R³ 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}

 

[lippyka]

Thanks!The matrix [T]\alpha\alpha is the matrix representation of the linear transformation T with respect to the basis \alpha. You can calculate it by expressing the vectors in R3 in terms of the basis \alpha. For example, for the first column of the matrix, you would need to express the vector (1,0,0) = 1*(3,0,1)+0*(3,1,1)+0*(2,1,1). This gives you the first column of the matrix as (1, 0, 0). Similarly, you can calculate the other columns.