Linear Algebra Questionproofs theorems preferably dont use

Linear Algebra Question,proofs / theorems preferably dont use DE

Prove: The space spanned by two vectors in R^3 is a line through the origin, a plane through the origin, or the origin itself.

Solution

span{u,v} certainly contains 0 since it contains v - v, thus the span of any pair of vectors (or even any singleton vector) must contain the origin.

Now note that if u = v = 0, then this span is exactly 0 and nothing more (this is a zero dimensional vector space).

Suppose u and v are not both 0. Then u and v are either linearly independent, or they aren\'t. Two vectors are linearly independent if and only if one is a scalar multiple of the other. If this is the case, then their span can only be a single line (through the origin ofcourse).

If they aren\'t scalar multiples of each other, then you get a unique plane that contains both vectors since their span contains all multiples of both and all linear combinations of multiples of both.

These cases are exhaustive, thus one of the 3 above statements is true.