---

Dimensionality check: Base case

Check the forward propagation dimensions for

1. the input layer \(A^{[0]}\)
2. the first hidden layer \(A^{[1]}\)

\[\begin{align*}
  \tag{1}
  A^{[0]}_{p^{[0]} \times n^{[0]}}
  & = A^{[0]}_{n \times m} = X_{n \times m}
  \\
  \tag{2}
  \boxed{A^{[1]}_{p^{[1]} \times n^{[1]}}}
  & \stackrel{?}{=}
  \boxed{A^{[1]}_{p^{[1]} \times p^{[0]}}}
\end{align*}
\]

with

\[\begin{align*}
  p^{[0]} & = n^{[0]} \\
  p^{[1]} & = n^{[1]}
\end{align*}
\]

by

\[\begin{align*}
  \boxed{A^{[1]}_{p^{[1]} \times n^{[1]}}}
  & =
  g^{[1]}
  \big(
  W^{[1]}_{p^{[1]} \times n^{[1]}}
  A^{[0]}_{p^{[0]} \times n^{[0]}}
  + \vec{b}^{[1]}_{p^{[1]} \times 1}
  \big)
  && \text{by definition of \(A^{[i]}\)}
  \\
  & =
  g^{[1]}
  \big(
  W^{[1]}_{p^{[1]} \times p^{[0]}}
  A^{[0]}_{p^{[0]} \times n^{[0]}}
  + \vec{b}^{[1]}_{p^{[1]} \times 1}
  \big)
  && \text{substitute \(n^{[1]}\) with \(p^{[0]}\)}
  \\
  & =
  g^{[1]}
  \big(
  T^{[1]}_{p^{[1]} \times n^{[0]}}
  + \vec{b}^{[1]}_{p^{[1]} \times 1}
  \big)
  && \text{multiply \(W^{[1]} A^{[0]}\)}
  \\
  & =
  g^{[1]}
  \big(
  U^{[1]}_{p^{[1]} \times n^{[0]}}
  \big)
  && \text{add \(T^{[1]} + \vec{b}^{[1]}\)}
  \\
  & =
  g^{[1]}
  \big(
  U^{[1]}_{p^{[1]} \times p^{[0]}}
  \big)
  && \text{substitute \(n^{[0]}\) with \(p^{[0]}\)}
  \\
  & =
  \boxed{A^{[1]}_{p^{[1]} \times p^{[0]}}}
  && \text{apply \(g^{[1]}\).}
\end{align*}
\]