# Plan

- 2.2
- 2.3
- Learn whatever we don’t cover.

- Crash course on matrix representations from
*F*^{n} to *F*^{n}.
- First we need to represent all vectors by vectors.
- Define ordered basis for any vector space. Both finite and infinite dimensional.
- We can write vectors with respect to this ordered basis and come up with the coordinate vector.
- Consider 4
*x*^{3} + 2*x*^{2} + *x* + 1 = 43 + 57( − 2 + *x*) + 26( − 2 + *x*)^{2} + 4( − 2 + *x*)^{3} in *P*_{3}(*R*) with respect to the 2 basis.
- If {
*v*_{j}} is a basis for *B*, then *T*(*v*_{j}) = ∑_{i = 1m}*a*_{ij}*w*_{i}. The matrix is then *A* = (*a**i**j*).
- Matrix representation of linear transformations. Lower bracket is domain and upper is codomain.
- The space of linear transformations is a vector space.
- The bracket operation is a linear transformation.

# 2.3 Compositiion

- The composition of linear transformations is linear.
- Whenever compostion makes sense:
- $T(U\sb{1} + U\sb{2}) = TU\sb{1} + TU\sb{2}$ and (
*U*_{1} + *U*_{2})*T* = *U*_{1} + *U*_{2}*T*.
*T*(*U*_{1}*U*_{2}) = (*T**U*_{1})*U*_{2}.
*T**I* = *I**T* = *T*
*a*(*U*_{1}*U*_{2}) = (*a**U*_{1})*U*_{2} = *U*_{1}(*a*_{U2}).

- These also holds for matrices.
- proof of matrix multiplication. In particular, the bracket is multiplicative.