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Basis and dimension
To describe every vector in a space efficiently, we want enough vectors to generate the whole space but no redundant ones. A basis does exactly that.
A set of vectors is a basis when it is both:
- linearly independent;
- spanning.
Coordinate bases
In $\mathbb R^2$, the standard vectors
$$\mathbf e_1=(1,0),\qquad \mathbf e_2=(0,1)$$
form a basis. Every vector $(x,y)$ has the unique representation
$$(x,y)=x\mathbf e_1+y\mathbf e_2.$$
Other pairs of nonparallel vectors also form bases of $\mathbb R^2$.
Uniqueness
Spanning guarantees that every vector can be represented. Linear independence guarantees that the representation is unique.
If a spanning set contains redundant vectors, different coefficient choices may represent the same vector. Removing the redundancy can produce a basis.
Dimension
The dimension of a space is the number of vectors in any basis of that space. Thus
$$\dim(\mathbb R^2)=2,\qquad \dim(\mathbb R^3)=3.$$
The deeper reason that every basis has the same size will become part of the abstract theory of vector spaces. At this stage, basis and dimension provide a compact way to describe the independent directions available in familiar coordinate spaces.