Finite representation of subspaces
A common theme is the extraction of a set of linearly independent vector from a subspace that contains enough information to represent the subspace as a whole.
That is the job of "basis". We will define this concept more precisely.
A by-product is the concept of "dimension".
Verify that the set \[ \left\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix} \,,\, \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right\} \] is linearly independent. Also verify that this set spans $\mathbb{R}^2$.
That is, this subset of $\mathbb{R}^2$ is both linear independent and spanning.
These properties make it very special. In particular, every vector in $\mathbb{R}^2$ can be written as a linear combination of this set. E.g., \[ \begin{bmatrix} 3 \\ 5 \end{bmatrix} = 3 \begin{bmatrix} 1 \\ 0 \end{bmatrix} + 5 \begin{bmatrix} 0 \\ 1 \end{bmatrix}. \] And this can be done in a unique way. (Must be 3 and 5; no other set of coefficients can make this work)
Show the two sets \[ \begin{aligned} \mathcal{B}_1 &= \left\{ \begin{bmatrix} 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 1 \\ 1 \end{bmatrix} \right\} & \mathcal{B}_2 &= \left\{ \begin{bmatrix} 1 \\ 2 \end{bmatrix} , \begin{bmatrix} 3 \\ 5 \end{bmatrix} \right\}. \end{aligned} \] are both linearly independent, and they are both spanning sets of $\mathbb{R}^2$.
They are examples of basis of $\mathbb{R}^2$.
Stated intuitively, a basis of a subspace is a subset of vectors in this subspace that is
We can push this intuitive idea of bases having just the "right size" even further.
Let $B$ be a basis of a subspace $V$ of $\mathbb{R}^n$, then
Implicit in this statement is the fact that any two basis for $V$ must be of the exact same size.
I.e., if $B_1$ and $B_2$ are both bases of a subspace $V$ of $\mathbb{R}^n$, then $| B_1 | = | B_2 |$. This fact is not completely obvious. Indeed, it is a part of the "Basis Theorem". One intuitive explanation of this important fact is that a basis has to be of just the "right size".
Exercise. Can you explain why this has to be true? (Try the special case of $n=2$)