$5\mathrm{.}1.$ Two$-$Dimensional Convolution as a Matrix$-$Vector Multiplication $(10$ points$).$

Consider a convolutional neural network layer consisting of a $(33)$ convolution kernel filtering a $(55)$ input image to produce a $(33)$ output image without zero$-$padding. This convolution operation can be written as a matrix$-$vector multiply of the form

widehat$(d)=Wx$

where $x$ is an $(251)-$element input vector that represents the input image in raster$-$scan $($row$-$by$-$row$)$ ordering, widehat$(d)$ is a $(91)-$element output vector that represents the output image in raster$-$scan $($row$-$by$-$row$)$ ordering, and $W$ is a matrix of dimensions $(925)$ that has $81$ non$-$zero entries that contain the elements of the $(33)$ convolution kernel, defined as

$W=\left[\begin{array}{ccc}{w}_{11}& w_{12}& w_{13}\\ w_{21}& w_{22}& w_{23}\\ w_{31}& w_{32}& w_{33}\end{array}\right]$

In the table below, write in the terms for the nonzero entries of $W$ that correspond to the coefficients of the convolution kernel $W.$ For your convenience, the entries of $x$ and of widehat$(d)$ are shown above and to the left, respectively, of the table.

$\backslash$table$[[$W$,x_{11},x_{12},x_{13},x_{14},x_{15},x_{21},x_{22},x_{23},x_{24},x_{25},x_{31},x_{32},x_{33},x_{34},x_{35},x_{41},x_{42},x_{43},x_{44},x_{45},x_{51},x_{52},x_{53},x_{54},x_{55}$