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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?

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  • $\begingroup$ Do you mean $P_n$ (often denoted $\Bbb R_n[x]$) is the set (vector space) of polynomial with degree less or equal to $n$? $\endgroup$
    – user63181
    Jun 30, 2014 at 9:56
  • $\begingroup$ 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 $p, q \in P_n$. $\endgroup$
    – afedder
    Jun 30, 2014 at 10:01
  • $\begingroup$ 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]. $\endgroup$
    – user160867
    Jun 30, 2014 at 10:04

2 Answers 2

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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.

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  • $\begingroup$ 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? $\endgroup$
    – user160867
    Jun 30, 2014 at 10:09
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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.

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  • $\begingroup$ 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? $\endgroup$
    – user160867
    Jun 30, 2014 at 10:14
  • $\begingroup$ 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. $\endgroup$
    – Adam Hughes
    Jun 30, 2014 at 10:15
  • $\begingroup$ 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? $\endgroup$
    – user160867
    Jun 30, 2014 at 10:17
  • $\begingroup$ 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. $\endgroup$
    – Adam Hughes
    Jun 30, 2014 at 10:19

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