1. Span and subspaces proof

Given that (a,b) and (c,d) are vectors in $R^2$ which do not lie on the same line through the origin then prove

1) the span Sp{(a, b), (c, d)} is equal to $R^2$
2) the proper subspaces of $R^2$ are the lines through the origin.

I think I'm pretty close with 1 by showing both sides of the equation are subsets of each other.

2. i don't remember linear algebra that well,

but isn't there a theorem saying that the span subspace of two
vectors that are not dependent on each other, equals the entire
vector space?

two vectors with two components each will span $R^2$
three vectors with three components each will span $R^3$

and by both vectors not lying on the same line,
you can gather that the are not co-dependent ---> those two vectors are a base of $R^2$

3. A set of vectors is a "basis" for vector space of dimension n if and only if any two of these are true:
1) There are n vectors in the set.
2) The vectors are independent
3) The vectors span the space

and then the third is also true.

Hopefully you know that $R^2$ has dimension 2 and there are two vectors in your set. All you need to do for (1) is show that they are independent.

For (2) you need to show that every straight line through the origin is a vector space (of what dimension?) and then that any set that contains two vectors NOT on the same straight line is not a subspace. You might make use of the fact that any line that goes through the origin can be written in parametric form as $x_1= a_1t$, $x_2= a_1t$, ..., $x_n= a_nt$, while for a line not through the origin there must be an added constant on at least one of those.