Suppose $XY$ is an $m $ by $n$ matrix, where $X$ is a $m$ by $k$ matrix and $Y$ is a $k$ by $n$ matrix. $y_i$ are the columns of $Y$ and $x_i$ are the columns of $X$. How do we know that $XY=\Sigma^{k}_{i=1}x_{i}y_{i}^{T}$ ?
I made an attempt to justify this.
$$XY = [x_1|x_2|..|x_k][y_1|y_2|..|y_n]$$
$$XY = X[y_1,y_2,..,y_n]=[Xy_1|Xy_2|..|Xy_n]$$
$$XY=[\Sigma_{i=1}^{k} y_{i1} x_{i}|\Sigma_{i=1}^{k} y_{i2} x_{i}|..|\Sigma_{i=1}^{k} y_{ik} x_{i}|0_{k+1}|...|0_n]$$
I am stuck here