I am unsure what does $T(v_i)$ means as shown in pictures below (I will type out the gist of it too)
Given a linear operator(transformation) $T$ on vector space $V$. Let a basis $B$ of $V$ be $B=\{v_1, v_2, ... , v_n\}$ where $n = \operatorname{dim} V$.
The matrix representation of $T_B$ (relative to basis $B$) $$T_B= \begin{pmatrix}a_{11}&a_{12}&\ldots&a_{1n}\\a_{21}&a_{22}&\ldots&a_{2n}\\\vdots&\vdots&\ddots&\vdots\\a_{n1}&a_{n2}&\ldots&a_{nn}\end{pmatrix}$$
So here is the part I do not understand: $$T(v_i) = a_{1i}v_1 + a_{2i}v_2 +...+a_{ni}v_n$$ This will give us the $i$th column of $T$
Is $T(v_i)$ a matrix multiplication of $T$ on $v_i$? Or does it simply refer to the $i$th column of T. The 2nd image I pasted shows my attempts at plugging in real values.
Also, just realised another obvious fact. If $T_B$$(v_i)$ = $a_{1i}v_i +a_{2i}v_2 + ... + a_{ni}v_n$, then $(a_{1i},a_{2i},...,a_{ni})$ should be the $i$th row instead of the $i$th column of $T_B$ since vectors are represented as columns and so the coefficients depend on the rows of the matrix to its left.
