AES Mix Column Transformation Section 3.4.3 "The MixColumns Step"(page 39) of "The Design of Rijndael", it states


  The columns of the state are considered as polynomials over
  $\operatorname{GF}(2^8)$ and multiplied modulo $x^4 + 1$ with a fixed polynomial $c(x)$.
  
  The polynomial $c(x)$ is given by
  $c(x) = \mathtt{03} \cdot x^3 + \mathtt{01} \cdot x^2 + \mathtt{01} \cdot x + \mathtt{02}$
  
  Let $b(x) \equiv c(x) \cdot a(x) \pmod{x^4 + 1}$. Then
  
  


How this polynomial multiplication modulo $x^4+1$ is converted to Matrix Multiplication?
The columns of the state are considered as polynomials over $\operatorname{GF}(2^8)$


Bytes are values in $\operatorname{GF}(2^8)$. Thus a column composed as the 4 bytes $a_0,a_1,a_2,a_3$ represent the polynomial  $a_0 x^0 + a_1 x^1 + a_2 x^2 + a_3 x^3$ (notice the ordering with respect to the power!).

Now we have:

\begin{array}{cccccccccccccccc}
(a_3 x^3 &amp;+&amp; a_2 x^2 &amp;+&amp; a_1 x^1 &amp;+&amp; a_0 x^0) &amp;\cdot&amp;\\
&amp; &amp; (\texttt{03}x^3 &amp;+&amp; \texttt{01}x^2 &amp;+&amp; \texttt{01}x^1 &amp;+&amp; \texttt{02}x^0) = \\
\texttt{03}a_3 x^6&amp;+&amp; \texttt{01}a_3 x^5 &amp;+&amp; \texttt{01}a_3 x^4 &amp;+&amp; \texttt{02}a_3 x^3 &amp;+&amp; \\
&amp; &amp; \texttt{03}a_2 x^5&amp;+&amp; \texttt{01}a_2 x^4 &amp;+&amp; \texttt{01}a_2 x^3 &amp;+&amp; \texttt{02}a_2 x^2 &amp;+&amp; \\
&amp; &amp; &amp; &amp; \texttt{03}a_1 x^4&amp;+&amp; \texttt{01}a_1 x^3 &amp;+&amp; \texttt{01}a_1 x^2 &amp;+&amp; \texttt{02}a_1 x^1 &amp;+&amp; \\
&amp; &amp; &amp; &amp; &amp; &amp; \texttt{03}a_0 x^3&amp;+&amp; \texttt{01}a_0 x^2 &amp;+&amp; \texttt{01}a_0 x^1 &amp;+&amp; \texttt{02}a_0 x^0)
\end{array}

If we apply $\mod (x^4 + 1)\ $ we have:
\begin{array}{cccccccccccccccc}
(a_3 x^3 &amp;+&amp; a_2 x^2 &amp;+&amp; a_1 x^1 &amp;+&amp; a_0 x^0) &amp;\cdot&amp;\\
&amp; &amp; (\texttt{03}x^3 &amp;+&amp; \texttt{01}x^2 &amp;+&amp; \texttt{01}x^1 &amp;+&amp; \texttt{02}x^0) = \\
\texttt{02}a_3 x^3 &amp;+&amp; \texttt{03}a_3 x^2 &amp;+&amp; \texttt{01}a_3 x^1 &amp;+&amp; \texttt{01}a_3 x^0 &amp;+&amp; \\
\texttt{01}a_2 x^3 &amp;+&amp; \texttt{02}a_2 x^2 &amp;+&amp; \texttt{03}a_2 x^1 &amp;+&amp; \texttt{01}a_2 x^0 &amp;+&amp; \\
\texttt{01}a_1 x^3 &amp;+&amp; \texttt{01}a_1 x^2 &amp;+&amp; \texttt{02}a_1 x^1 &amp;+&amp; \texttt{03}a_1 x^0 &amp;+&amp; \\
\texttt{03}a_0 x^3 &amp;+&amp; \texttt{01}a_0 x^2 &amp;+&amp; \texttt{01}a_0 x^1 &amp;+&amp; \texttt{02}a_0 x^0 &amp;= \\
b_3 x^3 &amp;+&amp; b_2 x^2 &amp;+&amp; b_1 x^1 &amp;+&amp; b_0 x^0
\end{array}

Thus by reading the colunns:

\begin{array}{cccccccccccccccc}
b_3 = \texttt{02}a_3 + \texttt{01}a_2 + \texttt{01}a_1 + \texttt{03}a_0\\
b_2 = \texttt{03}a_3 + \texttt{02}a_2 + \texttt{01}a_1 + \texttt{01}a_0\\
b_1 = \texttt{01}a_3 + \texttt{03}a_2 + \texttt{02}a_1 + \texttt{01}a_0\\
b_0 = \texttt{01}a_3 + \texttt{01}a_2 + \texttt{03}a_1 + \texttt{02}a_0\\
\end{array}

by reordering over $b$:

\begin{array}{cccccccccccccccc}
b_0 = \texttt{01}a_3 + \texttt{01}a_2 + \texttt{03}a_1 + \texttt{02}a_0\\
b_1 = \texttt{01}a_3 + \texttt{03}a_2 + \texttt{02}a_1 + \texttt{01}a_0\\
b_2 = \texttt{03}a_3 + \texttt{02}a_2 + \texttt{01}a_1 + \texttt{01}a_0\\
b_3 = \texttt{02}a_3 + \texttt{01}a_2 + \texttt{01}a_1 + \texttt{03}a_0\\
\end{array}

by reordering over $a$:

\begin{array}{cccccccccccccccc}
b_0 = \texttt{02}a_0 +  \texttt{03}a_1 + \texttt{01}a_2 + \texttt{01}a_3\\
b_1 = \texttt{01}a_0 +  \texttt{02}a_1 + \texttt{03}a_2 + \texttt{01}a_3\\
b_2 = \texttt{01}a_0 +  \texttt{01}a_1 + \texttt{02}a_2 + \texttt{03}a_3\\
b_3 = \texttt{03}a_0 +  \texttt{01}a_1 + \texttt{01}a_2 + \texttt{02}a_3\\
\end{array}

Which is exactly the matrix multiplication:

\begin{array}{cccccccccccccccc}
\begin{bmatrix}
b_0\\
b_1\\
b_2\\
b_3
\end{bmatrix}
=
\begin{bmatrix}
\texttt{02} &amp; \texttt{03} &amp; \texttt{01}&amp; \texttt{01}\\
\texttt{01} &amp; \texttt{02} &amp; \texttt{03}&amp; \texttt{01}\\
\texttt{01} &amp; \texttt{01} &amp; \texttt{02}&amp; \texttt{03}\\
\texttt{03} &amp; \texttt{01} &amp; \texttt{01}&amp; \texttt{02}
\end{bmatrix}
\times
\begin{bmatrix}
a_0\\
a_1\\
a_2\\
a_3
\end{bmatrix}
\end{array}