Why are polynomials infinite dimensional?

The vector space of polynomials in x with rational coefficients. Not every vector space is given by the span of a finite number of vectors. Such a vector space is said to be of infinite dimension or infinite dimensional.

How do you prove a polynomial is a vector space?

Let Pn be a set of all polynomials of degree n and smaller. Then, Pn is a vector space such that if p(x) E Pn then p(x) is uniquely represented by the basic functions {1, x, x2,…,x”}. Dimension of Pn is n +1.

What is the basis of an infinite dimensional vector space?

Infinitely dimensional spaces A space is infinitely dimensional, if it has no basis consisting of finitely many vectors. By Zorn Lemma (see here), every space has a basis, so an infinite dimensional space has a basis consisting of infinite number of vectors (sometimes even uncountable).

Can a subspace be infinite dimensional?

It is well known that all the subspaces of a finite dimensional vector space are finite dimensional. But it is not true in the case of infinite dimensional vector spaces. For example in the vector space C over Q, the subspace R is infinite dimensional, whereas the subspace Q is of dimension 1.

Is polynomial finite dimensional?

Thus every polynomial is expressible as a linear combination of the vectors in this set. But then which implies that is linearly independent. This is clearly false, hence a contradiction. Thus the vector space of polynomials is infinite dimensional.

Can a vector space be infinite dimensional if it is spanned?

A vector space is infinite-dimensional if it is spanned by an infinite set. By definition of basis: “A basis of a vector space V is a linearly independent subset of V that spans V.” The only three-dimensional subspace of R3 is R3 itself.

How do you prove the basis of a vector space?

Build a maximal linearly independent set adding one vector at a time. If the vector space V is trivial, it has the empty basis. If V = {0}, pick any vector v1 = 0. If v1 spans V, it is a basis.

Are all infinite dimensional vector spaces isomorphic?

Two vector spaces (over the same field) are isomorphic iff they have the same dimension – even if that dimension is infinite. Actually, in the high-dimensional case it’s even simpler: if V,W are infinite-dimensional vector spaces over a field F with dim(V),dim(W)≥|F|, then V≅W iff |V|=|W|.

How are functions infinite dimensional vectors?

Since the powers of x, x0= 1, x1= x, x2, x3, etc. are easily shown to be independent, it follows that no finite collection of functions can span the whole space and so the “vector space of all functions” is infinite dimensional.

Why is the set of all polynomials a vector space?

It happens that if you take the set of all polynomials together with addition of polynomials and multiplication of a polynomial with a number, the resulting structure satisfies these conditions. Therefore it is a vector space — that is all there is to it.

What is the definition of a mathematical vector space?

2 $\\begingroup$A mathematical vector space is defined abstractly – I suggest you look up vector spaces online or in a text book, and check the definition. One motivating factor for looking at vector spaces is that Euclidean space fits the definition, and you can do geometry using vectors.

What is the ring of polynomials with coefficients in a field?

The ring of polynomials with coefficients in a field is a vector space with basis 1, x, x2, x3, …. Every polynomial is a finite linear combination of the powers of x and if a linear combination of powers of x is 0 then all coefficients are zero (assuming x is an indeterminate, not a number).

Why do we look at vector spaces?

One motivating factor for looking at vector spaces is that Euclidean space fits the definition, and you can do geometry using vectors. But having abstracted the key idea from the Euclidean case, we find that there are lots of other objects which fit the definition – that’s one of the reasons it is so useful.