Fundamental Theorem of Matrices

Linear transformations between finite dimensional vector spaces behave exactly like matrices

1. If $M\in M_{n\times k}(\mathbb{R})$ then, $T_M$ is a linear transformation
2. If $T:V\to W$ has matrix $M=[T{]}_{\alpha }^{\beta }$ then $T=T_M$
3. If $M\in M_{n\times k}(\mathbb{R})$ then $M=[T_m{]}_{\alpha }^{\beta }$

Example Utilization

1. $T(x,y,z)=(x+y+z,y+x,z)$
2. $T(1,2,3)$ Solution:
3. $T(1,2,3)=[6,3,3]$ Verify other side
4. Find $[T{]}_{\alpha }^{\alpha }$
5. $T(1,0,0)=(1,1,0)$
6. $T(0,1,0)=(1,1,0)$
7. $T(0,0,1)=(0,0,1)$
8. Then, $$[T]_{\alpha}^{\alpha} = \left[\begin{array}{ccc} 1 & 1 & 1\ 1 & 1 & 0\ 0 & 0 & 1\ \end{array}\right]

6. Then, $$[T]_{\alpha}^{\alpha}[(1,2,3)] = \left[\begin{array}{ccc} 1 & 2 & 3\\ 1 & 2 & 0\\ 0 & 0 & 3\\ \end{array}\right] = [6,3,3]

Thus, they are equal