Trying to understand the Matrix approximation using SVD and watching ingular Value Decomposition (SVD): Matrix Approximation.
X is represented as the sum of $\sigma_i * u_i \cdot v_i^T$ where $u_i$ is the i-th column vector of $U$ and $v_i^T$ is the transposed i-th column vector of $V$.
It is stated in the video that this is because $\sigma_i * u_i$ is only multiplied by $v_i^T$ column, but I am confused why.
Question
If the operation between $U\cdot\Sigma$ and $V^T$ is the matrix multiplication, then each row element of $U$ as $u_(n)(row)$ should be multiplied by each column element of $V^T$ as in the red?
Why instead it becomes the multiplication between $u_i$ and $v_i^T$?
Also, the shape of $U$ is (n, n) and that of $V_T$ is (m, m) where n > m. Then how come a column $u_i$ of size n can be multiplied by a column $v_i^T$ of size m?