Important Property of Finite Dimensional Vector Spaces

• Let $V$ be a vector space with dimension $n$. Choose a basis $\{v_1,v_2,....,v_n\}$.
• Define $f:V\rightarrow \R^n$ by extending the function sending the basis vector $v_i$ to the standard basis vector $e_i\in \R^n$ for each $i$.
• Then f is an isomorphism.

The Matrix Corresponding to a Linear Transformation with respect to ordered bases.

• Let $f:V\rightarrow W$ be a linear transformation.
• Let $\beta =v_1,v_2,....,v_n$ be an ordered basis of $V$ and $\gamma =w_1,w_2,.....,w_m$ be an ordered basis of $W$.
• Each $f(v_i)$ can be uniquely written as a linear combination of $w_j$s, where $i=1,2,....,n$ and $j=1,2,....,m$

• Then the matrix corresponding to the linear transformation $f$ with respect to the ordered bases $\beta$ and $\gamma$ is given by:

Recovering the Linear Transformation

• Let $\beta=v_1,v_2,....,v_n$ and $\gamma=w_1,w_2,....,w_m$ be ordered bases of $V$ and $W$ respectively.

  

  

• Hence the matrix corresponding to $f$ is indeed $A$.

Fixed Ordered Bases: Linear Transformation $\leftrightarrow$ matrices

• Let $\beta$ and $\gamma$ be ordered bases for vector spaces $V$ and $W$ respectively where $n=dim(V)$ and $m=dim(W)$.