Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

how to get thier intersection?

share|improve this question
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 $$ [w_1,w_2,\ldots,w_m,u_1,\ldots,u_n]. $$

share|improve this answer

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.

share|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.