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Assuming we have 2 subspaces, $\mathbb W$ and $\mathbb U$ of $\mathbb V$.

how to get thier intersection?

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What do you mean by "get their intersection"? –  Taro Feb 28 '13 at 9:46

2 Answers 2

up vote 5 down vote accepted

Suppose that $\mathbb{W}=\operatorname{span}(w_1,w_2,\ldots,w_m)$ and $\mathbb{U}=\operatorname{span}(u_1,u_2,\ldots,u_n)$ where each of these is a minimum basis for the respective sets. If a vector $a$ is in the intersection of $\mathbb{W}$ and $\mathbb{U}$, it must be able to be expressed as a linear combination of the spanning set of $\mathbb{W}$ and $\mathbb{U}$. i.e. $$ a=c_1w_1+c_2w_2+\ldots+c_kw_m \text{ and } a=d_1u_1+d_2u_2+\ldots+d_ku_n $$ for some $c_i$ and $d_i$ to be determined. Since $a$ is arbitrary, you must have $$ c_1w_1+c_2w_2+\ldots+c_kw_m=d_1u_1+d_2u_2+\ldots+d_ku_n $$ which is equivalent to $$ c_1w_1+c_2w_2+\ldots+c_kw_m+e_1u_1+e_2u_2+\ldots+e_ku_n=0 $$ where $e_k=-d_k$. Since $a$ was an arbitrary element of $\mathbb{W}$ and $\mathbb{U}$, this system of linear equations must have a solution. Thus, the intersection of $\mathbb{W}$ and $\mathbb{U}$ is the nullspace of the matrix $$ [w_1,w_2,\ldots,w_m,u_1,\ldots,u_n]. $$

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If your subspaces are defined by two systems of homogeneous linear equation, just combine them in a single system, and the set of solutions will be the intersection.

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