Thread: Vector spaces, subspaces

I just recently learn about this subject and i don't have a single clue about it. Can anyone perhaps able to assist me in any way? An example and application of it would be nice.

What do you mean by "recently learn about this subject"? What did you learn? Linear algebra deals with finite dimensional vector spaces. Those are sets of objects in which we have some notion of "addition" that satisfies all the usual properities of addition and a "scalar multiplication" where we multiply vectors by numbers (more abstractly, any member of an algebraic "field"). One application is to solving systems of equations.
For example, if we are asked to solve
2x- 3y+ z= -6
x+ y+ z= -4
x- y- z= -2

We can set it up as a "matrix" equation, separating the coefficients from the unknowns:

and then there are a number of ways to solve that- we could find the inverse matrix and multiply both sides of it (equivalent to solving "Ax= b" by dividing both sides by A:
or, simpler than finding the inverse, from a purely computational point of view, write the "augmented matrix"

and "row reduce".

Linear algebra can also be applied to linear differential equations. The set of all solutions to an nth order differential equation is a vector space of dimension n.