What is a conservative vector field?

Questions:
Vectors, Matrices and Vector Calculus:

1. How are vectors and matrices related?
2. Can matrices move vectors around? If yes, how?
3. What dimensions do cross products exist on?
4. What is an Eigen vector? Why is it special?
5. Give an example of a vector space of 3 dimensions.
6. What dimensions does a M2x2 () have? Give an example.
7. Why do we need to diagonalize a matrix?
8. What is the Gram-Schmitt Process?
9. What is a vector field?
10. What is a conservative vector field?