I feel so lost in this section regarding vector spaces, sub spaces, spanning sets and basis.
I understand the basic concepts regarding vectors where basically a vector gives magnitude and direction. Multiplying a scalar by a vector basically scales the vector by the amount multiplied, and adding vectors is a matter of adding the components of the vectors to give a new vector.
I don't understand how all of this ties to the above examples. I'm working problems and find myself kind of understanding what I'm doing, but then I hit a problem like the one below:
Determine whether the set $S$ spans $R^2$. If the set does not span $R^2$, then give a geometric description of the subspace that it does span.
I created a linear combination of these vectors and found that they produce an infinite amount of solutions, but I don't know what to do with this information nor what it means.
I think this is because my underlying conceptual understanding is not strong. Can someone help me to understand what is happening in all of this?