I'm trying to solve this problem, but at the end I find something's wrong with my work. Here is the problem:
We're given the bases: $$ \beta = \bigg\{\begin{pmatrix}1\\1\end{pmatrix} \begin{pmatrix}1\\-1\end{pmatrix}\bigg\} \\ \gamma = \text{Canonical for }\mathbb{R}^4 $$
We're given the transformation from the vectores of the basis to the image: $$ T((1,1)) = \begin{pmatrix}1\\0\\0\\1\end{pmatrix} \\ T((1,-1)) = \begin{pmatrix}0\\1\\-1\\0\end{pmatrix} $$
And were asked to:
- Find $[T]_\beta^\gamma$
- If $x = (5,1)$, find $[x]_\beta$
- Determine $[T(x)]_\gamma$
- Find $T(x)$
The first part I think is simple, since all we need to do is find the transformation of the base for $\beta$, but it is already given, and since $\gamma$ is the canonical, the transformations should be $[T]_\beta^\gamma$.
The second part is the application of the definition, so we find that $(5,1)$ may be represented by the coordinate vector $(3,2)$.
Then, for number three, we use the theorem that states that $[T(x)]_\gamma = [T]_\beta^\gamma[x]_\beta$. That is $(3,2,-2,3)$.
Finally, we use the coordinate vector and use the entries as coefficients for $\gamma$ to find $T(x)$. This is $(3,2,-2,3)$, since $\gamma$ is the canonical basis.
The problem is that if I try to multiply $[T]_\beta^\gamma$ by the vectors of $\beta$, I don't get the transformation I'm expecting, according to what is given. What am I doing something wrong?
With the help of ᛥᛥᛥ:
I was thinking of something else. The process here is correct.
My problem was that I was expecting $[T]_\beta^\gamma$ to do what $[I]_C^\beta$ does, which is to transform a vector $x$ to the image, or do what $T(x)$ does. $[I]_C^\beta$ is the matrix formed by the transform of the canonical basis of the domain with $T(x)$.