Dimension of weight matrix in neural network
There seems to be an error in the screenshot. The weight, $W$ should be transposed, please correct me if I am wrong.


You are wrong.

Matrix multiplication works so that if you multiply two matrices together, $C = AB$, where $A$ is an $i \times j$ matrix and $B$ is a $j \times k$ matrix, then C will be a $i \times k$ matrix. Note that $A$'s column count must equal $B$'s row count ($j$).

In the neural network, $a^{[1]}$ is a $n^{[1]} \times 1$ matrix (column vector), and $z^{[2]}$ needs to be a $n^{[2]} \times 1$ matrix, to match number of neurons.

Therefore $W^{[2]}$ has to have dimensions $n^{[2]} \times n^{[1]}$ in order to generate an $n^{[2]} \times 1$ matrix from $W^{[2]}a^{[1]}$