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!

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.

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