Conjugate vectors What are conjugate vectors?
Can I have an example of it?
[ This question is in respect to finding the roots of equations with conjugate direction methods]
 A: 
What are conjugate vectors?

I was confused by this too. The word conjugate is quite an overloaded one in Math. The plain English definition of the word is hard enough to follow and seems to bear little intuitive relationship to the various Mathematical concepts. Anyway:
The concept of conjugate vectors is closely related to orthogonal vectors and linearly independence of vectors. However a conjugates are always defined w.r.t. some positive definite square Matrix:
Let $u,v$ be vectors in $R^n$ and let $A$ be a positive definite $n\times n$ matrix. $u$ and $v$ are said to be mutually $A$-conjugate if and only if $u^TAv = 0$ 1.
If $u,v$ are conjugate vectors any two vectors parallel to $u$ and $v$ respectively are also conjugate. So you'll often hear speak of conjugate directions rather than vectors as the scale doesn't matter. Also, any set of mutually $X$-conjugate vectors for some positive definite $n\times n$ matrix $X$ is also linearly independent. So in the rough, you can think if conjugate vectors as a stricter form of linearly independent vectors.

Can I have an example of it?

$u = (1,1), v = (1,-1) \text{ w.r.t the identity matrix } I_2$
A: If you only are interested in roots of polynomials (but then I can't understand why the "vectors" thing...), then all the roots of some given, fixed irreducible polynomial over some given field are called conjugate (elements, roots, etc.)
