Is it possible to remove a single element from a linear subspace without affecting the dimension? More formally, is there any example where the following is possible?
Given a linear subspace $\mathbf{A}$ and an element $\mathbf{x}\in\mathbf{A}$ there exists another linear subspace $\mathbf{B}$ such that $\mathbf{x}\notin\mathbf{A}\cap\mathbf{B}$, yet $\dim\mathbf{A}=\dim\mathbf{A}\cap\mathbf{B}$.
Intuitively this seems trivially impossible, but I'm having trouble seeing this clearly.

This cannot happen in finite dimension because
$$
\dim (A) = \dim (A\cap B)
$$
and
$$
A \supseteq A\cap B
$$
implies $A=A\cap B$. Since $x \in A$, we cannot have $x \notin A\cap B$.
