# Creation of a matrix representation of a linear transformation

Why is it that when finding the matrix representation of a linear transformation $$T: V \rightarrow W$$ where $$V$$ has basis $$\{v_1, v_2, ..., v_n\}$$ and $$W$$ has basis $$\{w_1,w_2,...,w_m\}$$, the resulting matrix sometimes has columns $$T(v_1), T(v_2),...,T(v_n)$$ and other times it has columns representing the coefficients of the linear transformation required to map the $$w_k$$ onto each $$T(V_k)$$?

• Can you provide an example of the latter? If I’m correctly understanding what you mean, the two are in fact the same. – amd Jul 14 at 6:10

Let $$\textsf{T}: \textsf{V}\to \textsf{W}$$ be a linear transformation and let $$\beta =\{v_1,v_2,\dots,v_n\}$$ and $$\gamma =\{w_1,w_2,\dots,w_m\}$$ basis for $$\textsf{V}$$ and $$\textsf{W}$$, respectively.
To compute the matrix associated with $$\textsf{T}$$ respect to $$\beta$$ and $$\gamma$$, we always, that is to say always, obtain the vector $$\textsf{T}(v_j)$$ and put them as a linear combination of the elements that conform $$\gamma$$, like this : $$\textsf{T}(v_j)=A_{1j}w_1+A_{2j}w_2+\cdots+A_{mj}w_m$$ then, we put the coefficients $$\begin{pmatrix} A_{1j} \\ A_{2j} \\ \vdots \\ A_{mj} \end{pmatrix}$$ in the $$j$$-th column of the matrix $$A:= [\textsf{T}]_{\beta}^{\gamma}$$ (for $$j=1,2,\dots,n$$).
Now, for your specific doubt, consider $$\textsf{T}: \mathbb{R}^2 \to \mathbb{R}^2$$ given by $$\textsf{T}(a,b)=(-b,a)$$ (doesn't matter) and the basis $$\beta=\{(2,1),(3,5)\}$$ and $$\gamma=\{e_1,e_2\}$$ (the standard basis).
Above I said, we must first to compute $$\textsf{T}(2,1)$$ and the result put it as a linear combination of the elements in the second basis. In this case $$\textsf{T}(2,1)=(-1,2)=(-1)e_1+2e_2$$ As you can see, the coefficients that we will put in the first column of the matrix will look exactly like the vector $$(-1,2)$$ (maybe this is what you meant). To finish, note also that $$\textsf{T}(3,5)=(-5,3)=(-5)e_1+3e_2$$ So $$[\textsf{T}]_{\beta}^{\gamma} =\begin{pmatrix} -1 & -5 \\ 2 & 3 \end{pmatrix}$$