I am a bit confused with span, basis, and dimension (when dealing with vector spaces).
My teacher told us that a span is a finite linear combination. And I know that a basis is a spanning, linearly independent subset, and the dimension is basically the cardinality of a basis.
MY question is, how can a dimension be infinite dimensional? Isn't a basis supposed to be finite because a span is finite?
I know I must be misinterpreting something, can anyone help clarify?
Thank You.
Best Answer
You have to be careful with your wording. Saying "a span is finite" doesn't really mean anything. The span of a collection of vectors is the set of all finite linear combinations of those vectors.
Consider the vector space of all real polynomials $\mathcal{P}(\mathbb{R})$. It has a basis $\{x^n \mid n \in \mathbb{N}\cup\{0\}\}$ which has infinite cardinality, so $\mathcal{P}(\mathbb{R})$ is infinite dimensional. Any finite linear combination of these polynomials gives you an element of $\mathcal{P}(\mathbb{R})$. If you were to take a finite collection of the polynomials, say $\{x^{n_i} \mid i =1, \dots, k\}$, then their span would not contain any polynomial of degree more than $\max_i n_i$, so you would not obtain $\mathcal{P}(\mathbb{R})$.