Let $A, B\in \mathbb{R^{m \times b}}$. Show if $A\vec{e_i} = Be_i $ $\forall i\in \{1, ..., n\}$ then $A=B$.

Question

Note: $\vec{e_i}$ is the $i$th column vector of the identity matrix.

Relevant theorem: $A\vec{x} = x_1\vec{a_1} + x_2\vec{a_2} + ... + x_n\vec{a_n}$ This states that a matrix times a vector is each component of the vector times the appropriate column of the matrix. We want to leverage this formula where the vector is a column vector of the identity matrix, but I'm not sure how.