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Matrix of Linear Transformations Determines Elements of Kernel

Let $V$ be a finite dimensional vector space over the field $F$ and suppose $T$ is a linear transformation from $V$ to $V$. Let $W=text{ker}(T)$. Prove that if $T(v)$=0 for every $v$ in $V$, then the matrix of $T$ in the basis ${v_1, v_2, cdots v_n}$ determines the elements of $W$.
My thoughts: Let $w$ be an arbitrary vector in $W$, say $w=Tv_1+Tv_2+cdots+Tv_n$. If $T(v_i)=0$, then $T(w)=T(Tv_i)=0$, so $w$=0. Thus, $T$ satisfies the condition that $T(v)$=0 for all $v$ in $V$. Therefore, the elements of $W$ are uniquely determined by the matrix of $T$. But how do we actually see what elements of $W$ are taken care of in this example, specifically $w$, without seeing the basis? (Bear in mind this is not homework, I am teaching myself linear algebra)

A:

For a matrix, a kernel element is an element $w = (w_1,dots,w_n)$ with $Aw = 0_n$ (where $0_n$ is the zero vector in $mathbb F^n$).
In your example, you know that $W = ker T$, so the space $W$ is given

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