My linear algebra class went from 0-100 real quick. I've attended every single lecture (so I know I haven't missed out on anything); however, very recently he has been using the notation $L_A$ for a linear transformation.
This is the most understandable definition I have found for $L_A$:
- Definition: Let $A$ be an $m × n$ matrix. Then we define the linear transformation $L_A : R^n → R^m$by the rule $L_Ax := Ax$ for all $x ∈ R^n$, where we think of the vectors in $R^n$ and $R^m$ as column vectors.
but I still am having a hard time understanding what it really means. Any clarification would be appreciated!
Thank you.
Given a linear transformation $L:V\to W$, we associate a matrix $A_L$ by choosing bases $v_k,w_l$ for $V$ and $W$, and letting $A_L=(a_{ij})$ where $a_{ij}$ is the coefficient of $w_i$ in the expansion of $L(v_j)$, $$ L(v_i)=\sum_{j}a_{ij}w_j. $$ Conversely, a matrix of numbers $A$ determines a linear transformaion $L_A:V\to W$ once bases of $V,W$ have been chosen.
What makes this correspondence possible is that a linear map (a function $L$ between vector spaces satisfying $L(av_1+v_2)=aL(v_1)+L(v_2)$) is determined uniquely by its values on a basis.