geometric description of subsets that form bases in $\mathbb R^2$ and $\mathbb R^3$ How do you give a geometric description of the subsets of $B$ in $\mathbb R^2$ and $\mathbb R^3$ that form bases?  would the bases of the space have the same number of points.  I'm having difficulty understanding bases and forming them.
 A: Hint for part 1. A basis of a $k$-dimensional vector space $V$ is a set of vectors $B=\{\mathbf{v}_1,\dots,\mathbf{v}_k\}$ such that you can write any $\mathbf{v}\in V$ as a linear combination of vectors in $B$. Can you think of a simple, obvious example of such a set of vectors for $\mathbb{R}^2$? What would be a geometric description of that set? Can you generalize from that?
As for the second part, I'm not sure what you mean... Are you asking if the bases contain the same number of vectors?
A: Let $V$ be a linear space (aka vector space).  Then a basis for $V$ is a subset of points $\{v_i\}_{i \in I}$, such that each vector in $V$ can be written in one and only one way as a finite linear combination of vectors from $\{v_i\}$.
As for a geometric description in $\mathbb{R}^3$ all bases have $3$ elements, so basically think of a set of $3$ vectors.  To form a basis for $\mathbb{R}^3$, the vectors need to form edges of a parallelopiped (geometric shape) with one corner at the origin, that has nonzero volume.
Now prove all the above.  
A: A subset $B$ of ${\mathbb R}^2$ is a basis if it consists of exactly 2 (non-zero) vectors that are not colinear, i.e., that are not both on the same line.
A subset $B$ of ${\mathbb R}^3$ is a basis if it consists of exactly 3 (non-zero) vectors that are not coplanar, i.e., that are not all three on the same plane.
