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I'm quite confused with this theorem

Can someone explain me step by step how do I find a linear transformation from a given matrix and finding a matrix from a given linear transformation please? Examples will be great.

Thanks alot.

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    "Step by step" would be too weary to do here. For that there exist books, notes, tutors, etc. Here you can ask for some help in a specific point. – 2017-02-07
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    The easy direction is "from matrix to linear transformation." Do you understand how to do this part? – 2017-02-07

2 Answers 2

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Ok given a matrix Assuming it to be $ n \times n = (a_{ij})$ Now let V be a vector space of dimension n over a field where the matrix has its values. Now let $ v_1,v_2,..v_n$ be a basis of V. Define $ A : V \to V$ by $A(v_i) = \sum_{j=1}^n a_{ji} v_j$.

Conversely given a map from V to V. define a matrix whose entries are $ a_{ij}$ where $a_{ij}$ is the coefficient of $v_j$ in the expansion of $A(v_i)$in terms of $v_i$'s.

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Let $$A=\begin{pmatrix} 1 & 0 & 9 \\ 0 & -2 & 1 \\ 0 & 0 & 3 \end{pmatrix}\in\Bbb R^{3\times 3}$$ be a matrix. The linear map associated to it is the map \begin{align*} F_A:\Bbb R^3 &\longrightarrow \Bbb R^3 \\ \begin{pmatrix} a\\b\\c\end{pmatrix} &\longmapsto \begin{pmatrix} 1 & 0 & 9 \\ 0 & -2 & 1 \\ 0 & 0 & 3 \end{pmatrix} \cdot\begin{pmatrix} a\\b\\c\end{pmatrix} = \begin{pmatrix} a+9c\\-2b+c\\3c\end{pmatrix}. \end{align*} Conversely, assume you have any linear map $F:\Bbb R^3\to \Bbb R^3$. Let \begin{align*} e_1 &= \begin{pmatrix} 1\\0\\0\end{pmatrix}, & e_2 &= \begin{pmatrix} 0\\1\\0\end{pmatrix}, & e_3 &= \begin{pmatrix} 0\\0\\1\end{pmatrix}. \end{align*} Let $$ \begin{pmatrix} A_{1i}\\A_{2i}\\A_{3i}\end{pmatrix} := F(e_i), $$ this defines a Matrix $A\in\Bbb R^{3\times 3}$ with entries $A_{ij}$.

If you apply this to our $F_A$ above, you see how we get the matrix $A$ back. All of this generalizes as explained in the other answer.