Imagining basis vectors in XY coordinate system I have just started learning Linear Algebra and in Vectors there is this concept of standard basis vectors , î and j, and all the vectors can be expressed as the sum of these two basis vectors. I want to know if any two random vectors can also serve as basis vectors ? What is the intuition behind this ?
 A: The definition of a basis of a vector space is: a set of vectors belonging to the space that satisfy the following: any vector in the space can be represented as a unique linear combination (not just "the sum", which is one specific linear combination) of the vectors in the basis.
The underlying notion is linear independence. In order to be a basis for a finite dimensional vector space, a set of vectors must 1) span the vector space (so that any vector can be represented as a linear combination) and 2) must be linearly independent (so that the linear combination is unique).
Often people assume that a basis has to be orthonormal (mutually perpendicular, unit length vectors), but that's not the case (although such a basis does make the math simpler in many cases).
A: No, any two vectors are not always a basis. If you recall your definition of basis of a vector space from class, you will note that a set of vectors forms a basis if they possess two important properties:

*

*Linear independence

*They span the vector space

In the case of the XY plane, can you think of two vectors which violate one of these properties? Try starting with $\hat{i}$ and building another vector so that you still don't have a basis.
