Exercise 8.4

Suppose $Y=\left[\begin{array}{cccc}\hfill y^1\hfill & \hfill y^2\hfill & \hfill \dots \hfill & \hfill y^N\hfill \end{array}\right]$ and $X=\left[\begin{array}{cc}\hfill 1\hfill & \hfill x_1^T\hfill \\ \hfill \dots \hfill & \hfill \dots \hfill \\ \hfill 1\hfill & \hfill x_N^T\hfill \end{array}\right]$, wherethe columns of $Y$ is $y^n=\left[\begin{array}{c}\hfill 0\hfill \\ \hfill \dots \hfill \\ \hfill y_n\hfill \\ \hfill \dots \hfill \end{array}\right]$, the only non-zero entry is at the $n$-th row with value of $y_n$.

We use column times row in matrix multiplication, so

This is exactly the formula we are seeking for.