I'm taking a first course in Linear Algebra. I'm using Linear Algebra and Its Applications - 5th Ed. by David C. Lay. Chapter 5.4 (Eigenvectors and Linear Transformations) has been a tough chapter to grasp. This is a diagram I pulled from the textbook that shows what I'm talking about:
Also, here is a snapshot of my Professor's notes, which somehow made things even more confusing for me:
![[![Professor's Notes][2]][2])](https://i.stack.imgur.com/0ZSgI.jpg)
Can anyone describe what's going on here, or refer me to an alternate source for learning this chapter in particular?
Imagine a base $\mathcal{B} = (\mathbf{b}_1,\mathbf{b}_2,\cdots )$ for the space $V$, this means that given a vector $\mathbf{x}\in V$ we can always find a set of numbers $(r_1,r_2,\cdots)$ such that
$$ \mathbf{x} = r_1\mathbf{b}_1 + r_2\mathbf{b}_2 + \cdots $$
The array with all the $r$'s is called the coordinates of $\mathbf{x}$ in $\mathcal{B}$ and in the book you are following is denoted by $[\mathbf{x}]_\mathcal{B}$.
Same logic can be applied for a vector in the space $W$.
Now imagine a linear operator between $V$ and $W$, a bridge that allows you to transform the vector $\mathbf{x}\in V$ into a vector $\mathbf{y}\in W$, let us call that operator $T$
\begin{eqnarray} T: V &\to& W\\ \mathbf{x} &\mapsto& T(\mathbf{x}) = \mathbf{y} \end{eqnarray}
The question your teacher is trying to answer is how to find $\mathbf{y}$, or said in another words, what is the image of $\mathbf{x}$ under the transformation $T$, or how the coordinates will transform when you make the vector $\mathbf{x}$ go through the bridge. The answer is simple provided you know how to transform the vector of the basis under $T$, that is, if you know
$$ T(\mathbf{b}_{i}) $$
In this case
$$ T(\mathbf{x}) = T(r_1\mathbf{b}_1 + r_2\mathbf{b}_2 + \cdots ) = r_1 T(\mathbf{b}_1) + r_2T(\mathbf{b}_2) + \cdots $$