TITLE: Linear Algebra, Vector Space: how to find intersection of two subspaces?
QUESTION [11 upvotes]: $${ W = Sp\{{(1,3,4),(2,5,1)\}}\\ U = Sp\{{(1,1,2),(2,2,1)}} \}$$
Find a span $$U \cap W$$
First time using Math latex, pretty hard.

REPLY [6 votes]: Even though this is years after the question, posting following for completeness. This is very similar to solution by Pavan Karjol.
Let $P_u$ and $P_v$ be two projection matrices onto the two sub-spaces spanned by the columns of $U$ and $V$. These are calculated as $P_u=U(U^T U)^{-1} U^T$ and $P_v =V(V^T V)^{-1} V^T.$
Let $z$ be a vector that lies in intersection of these two sub-spaces. Since  $z$ is on both sub-spaces, projecting it onto each subspace doesn't change the vector.
Therefore, $z = P_uz$ and $z = P_vz$.
Substituting one in the other gives $z = P_uP_vz$ therefore, $0 = (P_uP_v-I)z$ .
Meaning, $z$ is in the null space of $M=(P_uP_v - I)$. Perform the eigenvalue decomposition on $M$ and select the eigenvectors corresponding to zero or nearly zero eigenvalues and that forms an orthogonal basis for the intersection of given sub-spaces.