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So I'm given 2 bases for the vector spaces $U$ and $V$. Suppose dim$(U \cap V) \geq2$, then how do I find a basis for it? Thanks in advance.

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    That depands on whats $U,V$...you need to work with the definition. thats what you wrote in the comment.2012-11-19

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Presumably $U,V$ are contained in a finite dimensional vector space $Z$. Let $W = U \cap V \subset Z$.

Let $u_1,...,u_{n_U}$ and $v_1,...,v_{n_V}$ be bases of $U,V$ respectively. Define $A: \mathbb{R}^{n_U} \to Z$ by $A x = \sum_k x_k u_k$ and similarly $B: \mathbb{R}^{n_V} \to Z$ as $B y = \sum_k y_k v_k$. Note that ${\cal R} A = U$, and ${\cal R} B = V$ and so $W = {\cal R} A \cap {\cal R} B$.

Define $\Gamma: \mathbb{R}^{n_U} \times \mathbb{R}^{n_V}\to Z$ by $\Gamma (x,y) = Ax-By$, and let $N = \ker \Gamma$. Let $\Pi_x : \mathbb{R}^{n_U} \times \mathbb{R}^{n_V}\to \mathbb{R}^{n_U}$ be the projection $\Pi_x (x,y) = x$.

Note that $\Gamma$ is given by the matrix $\Gamma = \begin{bmatrix} u_1 & \cdots & u_{n_U} & v_1 & \cdots v_{n_V} \end{bmatrix}$.

Then $p \in W$ $\iff$ there exists $x,y$ such that $Ax=By$ and $p=Ax$ $\iff$ there exists $x,y$ such that $\Gamma (x,y) = 0$ and $p=Ax$ $\iff$ $n \in N$ and $p=A\Pi_x n$ $\iff$ $p \in A \Pi_x N$.

So a procedure would be to find a basis $\nu_1,...,\nu_{n_N}$ for $N = \ker \Gamma$, and select a maximal linearly independent subset of $A \Pi_x \nu_1,...,A \Pi_x \nu_{n_N}$.

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    Yes, it involves finding a basis for $V^\bot$, say $v_1,...,v_k$. Then $x \in V$ $\iff$ $\begin{bmatrix} v_1^T \\ \cdots \\ v_k^T \end{bmatrix} x =0 $. Repeat for $U$ and stack one matrix on top of the other. Then any element in the stacked matrix null space must be in the null space of the original ones and vice versa.2012-11-20