All combinations of two vectors in $\mathbb{R}^3$ I am studying from materials provided by MIT for free and in one of the lectures, they give two vectors, u and v:
u = (1,-1, 0), v = (0, 1, -1) and they claim the following:
"The collection of all multiples of u forms a line through the origin. The collection of all multiples of v forms another line. The collection of all combinations of u and v forms a plane."
I think all possible combinations of the two vectors form (fill) the whole 3D space. Am I correct or is the material stating it correctly? Thanks
Link: https://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/ax-b-and-the-four-subspaces/an-overview-of-key-ideas/MIT18_06SCF11_Ses1.13sum.pdf
 A: The collection of all two of these line DO NOT fill all of $\mathbb{R}^3$. For example, what combination of $u$ and $v$  gives  the point $(2,0,-1)$? In general, if you have a set of $n$ vectors living in $\mathbb{R}^{n+k}$, then the subspace of $\mathbb{R}^{n+k}$ obtained by taking all linear combinations of those $n$ vectors is AT MOST an n-dimensional.
Remember when you were finding the normal vector to a plane back in calculus $3$? You did this by taking the cross product of any two non-parallel vectors in that plane. It's interesting that knowing only two vectors in the plane is enough to completely describe how the rest of the plane lies in $\mathbb{R}^3$! (or more generally in $\mathbb{R}^n$). In calculus three you learned that this was because given any two non-parallel vectors in a plane, the vector obtained from taking their cross product will be in the same direction, that is, the direction of the normal vector of that plane. When you get to linear algebra, you will learn that another reason that two non-parallel vectors in a plane completely determine the plane: The plane can be explicitly described as the set of all linear combinations of those two non-parallel vectors!!
These are some of the most very basic ideas in the subject of linear algebra. Linear algebra is the class that made me switch from a physics major to a math major back in the day. You're asking the right questions homey! Keep it up!
