Good day, I have a little doubt: It is well known that given two bases (or even one if we consider the canonical basis) of a vector space, every linear transformation $T:V \rightarrow W$ can be represented as a matrix, but since this is an isomorphism between $L(V,W)$ and $\mathbb{M}_{m\times n}$ where the latter represents the space $m\times n$ matrices on the same field in which are defined respectively vector spaces. That's where my question comes up, I know find the matrix associated with the linear transformation, but not know how to move from the matrix transformation, ie given any matrix, find the linear transformation that defines it. I wish you could please explain the theoretical process and to see a practical example. Thank you very much, I know it's definitely something silly, but I'm still a student.
[Math] How to find the linear transformation associated with a given matrix
linear algebra
Best Answer
The columns of the matrix tell us where the basis vectors of the domain are mapped, in terms of the basis vectors of the codomain. Since every vector in the domainis a linear combination of the basis vectors (in a unique way), we can extrapolate, in a sense, the image of any given vector. Let $A$ be an $m\times n$ matrix (with coefficients in a field $F$) with columns $A_1, ..., A_n$. Let $V$ be an $n-$dimensional $F-$vector space, and $W$ an $m-$dimensional $F-$vector space, with ordered bases $(v_1, ..., v_n)$ and $(w_1, ..., w_m)$, respectively. Finally, let $T$ be the linear transformation associated with $A$, and let $v\in V$ with $v = c_1v_1 + ... + c_n v_n$ (remember, this expression for $v$ as a linear combination of basis vectors is unique). Then $$T(v) = T(c_1v_1+...+c_nv_n) = c_1T(v_1)+...+c_nT(v_n) = c_1A_1 + ... + c_nA_n$$ So, this is how the matrix lets us calculate the image of any vector. Notice that the expression on the right is just the matrix $A$ multiplied by the vector $[c_1, c_2, ..., c_n]^T$.
For example, if we let $V = W = \Bbb{R}^2$ (considered as $\Bbb{R}-$vector spaces) with the standard basis, let $$A = \left[ \begin{array}{ccc} 1 & 2 \\\ 2 & 3 \end{array} \right]$$ And let $T$ be the linear transformation associated with $A$, and let $v = [1,5]^T$. Then $$T(v) = 1\cdot T(e_1) + 5\cdot T(e_2) = [1,2]^T+5[2,3]^T = [11,17]^T = A\cdot v$$