# How to see a matrix presents a linear transformation?

Consider the transformation $T:P_n \rightarrow P_n$ ($P_n$ is the vector space of polynomials of degree at most $n$, with complex coefficients) and its associated matrix presentation, namely $F$. How can I show that the transformation $T$ corresponding to $F$ is linear?

Generally, how to see a matrix presents a linear transformation?

• Do you mean $P_n$ (often denoted $\Bbb R_n[x]$) is the set (vector space) of polynomial with degree less or equal to $n$?
– user63181
Jun 30, 2014 at 9:56
• Recall the definition of linearity for a transformation $T$. If $T$ is linear, then $T(ap + bq) = aT(p) + bT(q)$ for constants $a,b$ and polynomials $p, q \in P_n$. Jun 30, 2014 at 10:01
• That is right. P_n is the set (vector space) of polynomial with degree less or equal to n with complex coefficients, denoted by C_n[x]. Jun 30, 2014 at 10:04

By the rules of matrix multiplication, we have $A(v+w)=Av+Aw$ and $Acv=cAv$ for $v,w\in \mathbb R^n$, $c\in\mathbb R$ (or whatever field). Hence $v\mapsto Av$ is linear.

• Thank you very much for the response. Does this rule works for if the matrix is infinite dimensional? Clearly, if the matrix A satisfies A(v+w)=Av+Aw and Acv=cAv, can we conclude that the transformation T associated to A is linear? Another question, can we have matrices corresponding to nonlinear transformation at all? Jun 30, 2014 at 10:09

A matrix is a way of presenting a group of linear forms all together.

If you write $A= \begin{pmatrix} - & v_1 & - \\ - & v_2 & - \\ \vdots & \vdots & \vdots \\ - & v_n & - \end{pmatrix}$

Then $Av=\begin{pmatrix} v_1\cdot v \\ \vdots \\ v_n\cdot v\end{pmatrix}_i$ by definition of matrix multiplication. But then you can easily see dot products are individually linear, and the rest is history.

• Thanks for your response. Can you please introduce me a book or a paper to see this problem from the point of view that you mentioned? Can we generalize this to infinite dimensional? Jun 30, 2014 at 10:14
• It's the definition of matrix multiplication and the definition of the dot product. Both of them involve taking a string, multiplying the corresponding elements, and adding them up. If you compare the two definitions you will see that they match up. Jun 30, 2014 at 10:15
• I want to have some more information about the idea, particularly for the space of polynomials. Can you please introduce me good references for this topic please? Jun 30, 2014 at 10:17
• Matrices represent linear transformations only after you use the coordinate map to identify $\mathbb{F}_n[x]$ with $\mathbb{F}^{n+1}$, where $\mathbb{F}$ is your underlying field, so it only needs to work on $\mathbb{R}^n$ where it is easy. Jun 30, 2014 at 10:19