How to construct a vector perpendicular to a bunch of known vectors?

[jollage]

Hi,

Given several vectors, which may be or not be orthogonal to each other, how to construct a vector perpendicular to them? In a sense of inner production being zero.

To be specific, I have $n$ vectors $v_{N}$ of length $N$, where $n<N$. So the maximum rank for these vectors is $n$, which leaves space for new vectors perpendicular to all of them. How to construct such a vector? I know Gram-Schmidt process, but it seems it's not what I want.

Thanks.

 

[micromass]

Why isn't the Gram-Schmidt process what you want?

 

[WWGD]

Take any vector in the orthogonal complement of the span of {v_1,v_2,..,v_n}. You can use, e.g., the fundamental theorem of linear algebra:

http://en.wikipedia.org/wiki/Fundamental_theorem_of_linear_algebra

to construct the ortho subspace.

 

[jollage]

Thanks. I see. I should use Gram-Schmidt process.

 

[WWGD]

You can, but you can also write a matrix M using your vectors, row-reduce, calculate the nullspace of M and then use the fact , by the fundamental theorem of algebrar ,that the nullspace of the matrix is the ortho complement of the row space, and then you can find a basis for the nullspace. Just an alternative.