Intersection of two subspaces in 4D I would like to know if there is some way to imagine the case when a 3D subspace intersects with a 2D plane in a 4D space.
For example, let's have a 3D space in 4D
$$A = \left(\begin{array}{c}1 & 2 & 3 & 4\\7 & 9 & 0 & 5\\ 3 & 3 & 4 & 1\end{array}\right)$$
and
$$B=\left(\begin{array}{c}1 & 1 & 0 & 4\\1 & 2 & 0 & 1\end{array}\right)$$
And if I am right, the basis of the space intersection is $C\approx(\begin{array}{c}1 & 1.1614 & 1.2248 & 2.1628)\end{array}$, which is a line.
My stupid question is. Is it possible that an intersection in 4D of a 3D space with a 2D plane is a 1D line? I can't picture how it is possible (it's like a 2D plane is "touching" a 3D space in one line?). 
 A: Let $e_1, e_2, e_3, e_4$ be the standard basis vectors of $\mathbb R^4$. The $3$-dimensional subspace
$$
U = \operatorname{span}(e_1, e_2, e_4) = \left\{ (x,y,0,w) \in \mathbb R^4 \right\}
$$
and the $2$-dimensional subspace
$$
V = \operatorname{span}(e_3, e_4) = \left\{ (0,0,z,w) \in \mathbb R^4 \right\}
$$
intersect at the line
$$
U \cap V = \operatorname{span}(e_4) = \left\{ (0,0,0,w) \in \mathbb R^4 \right\}.
$$
The situation here is similar to the situation of the $z$-axis intersecting the $xy$-plane at a point in $\mathbb R^3$. Here the $zw$-plane is intersecting the $xyw$-hyperplane at the $w$-axis.
A: The key result you need is that for a finite-dimensional vector space $V$, with subspaces $A,B$, we have $$\dim(A)+\dim(B)=\dim(A+B)+\dim(A\cap B)$$
Hence $5=\dim(A+B)+\dim(A\cap B)$.  Since $B\subseteq A+B$, we have $\dim(A+B)\in\{3,4\}$.  If $3$, then $A\cap B=A$.  If $4$, then $\dim(A\cap B)=1$.

Here's a related scenario to wrap your mind around.  In 3 dimensions, two planes can intersect in a plane or a line (or not at all, if they're parallel).  However, in 4 or more dimensions, two planes can intersect in a single point; specifically the origin if they are standard planes (not affine).  Here's how: one is $Span(e_1,e_2)$ and the other is $Span(e_3,e_4)$, where $\{e_1,e_2,e_3,e_4\}$ is the standard basis for $\mathbb{R}^4$.
