# How to quickly determine whether the intersection of two linear subspaces contain nonzero vectors?

by Wei Jiang   Last Updated November 18, 2018 09:20 AM

How to quickly determine whether the intersection of two linear subspaces contain nonzero vectors? Please give a algorithm.

i.e., both $$Ax=0$$ and $$Bx=0$$ have nonzero solutions. where the vector x has the same dimension. We know that all the solutions to $$Ax=0$$ is a linear subspace $$V_1$$, and all the solutions to $$Bx=0$$ is a linear subspace $$V_2$$. It is known that the intersection of $$V_1$$ and $$V_2$$ is also a linear subspace, which is denoted by $$V_3$$.

Please give a method to determine that whether $$V_3$$ has a nonzero vector. Thank you!

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#### Answers 1

Let $$M$$ be the matrix consisting of $$A$$ on top of $$B$$ (i.e. the rows of $$A$$, then the rows of $$B$$). The condition is that the rank of $$M$$ is less than its number of columns. Use row reduction to determine this.

Robert Israel
October 29, 2018 12:48 PM