Can a vector space be infinite dimensional?

Can a vector space be infinite dimensional?

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. We will now see an example of an infinite dimensional vector space.

Is R Infinity a vector space?

There are some vector spaces, such as R∞, where at least certain infinite sums make sense, and where every vector can be uniquely represented as an infinite linear combination of vectors.

Are functions infinite dimensional vectors?

Functions can be seen as infinite dimensional vectors. Basically the xth row in the vector gives the value for f(x).

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 vector have infinite elements?

Vectors can be defined over any field, using elements from that field, and can have length equal to an element of that field. Since the real numbers do not have any numbers of infinite size (since infinity is not itself a number), no vector made of real numbers will have infinite length.

Which of the following is vector space over R?

A vector space over R is a nonempty set V of objects, called vectors, on which are defined two operations, called addition + and multiplication by scalars · , satisfying the following properties: A1 (Closure of addition) For all u, v ∈ V,u + v is defined and u + v ∈ V .

What is the dimension of the vector space R over R Mcq?

Answer: We’ve just noted that R as a vector space over Q contains a set of linearly independent vectors of size n + 1, for any positive integer n. Hence R cannot have finite dimension as a vector space over Q. That is, R has infinite dimension as a vector space over Q.

How do you write infinity in R?

There are two types of infinity in R. The Inf and -Inf are positive and negative infinity, whereas NaN means ‘Not a Number’.

What is the dimension of a function space?

The zero vector is given by the constant function sending everything to the zero vector in V. The space of all functions from X to V is commonly denoted VX. If X is finite and V is finite-dimensional then VX has dimension |X|(dimV), otherwise the space is infinite-dimensional (uncountably so if X is infinite).

Which is not finite dimensional vector space?

For any field , the set of all sequences with values in is an infinite dimensional vector space. The space of continuous (or smooth, or whatever) functions on any non-empty real manifold of positive dimension is infinite dimensional.

Is R2 a vector space?

The vector space R2 is represented by the usual xy plane. Each vector v in R2 has two components. The word “space” asks us to think of all those vectors—the whole plane. Each vector gives the x and y coordinates of a point in the plane : v D .

What is a vector space over R?

The set of all real valued functions, F, on R with the usual function addition and scalarmultiplication is a vector space over R. The set of all polynomials with coefficients in R and having degree less than or equal ton, denoted Pn, is a vector space over R.

What are the components of the vector space R3?

The vectors have three components and they belong to R3. The plane P is a vector space inside R3. This illustrates one of the most fundamental ideas in linear algebra. The plane going through .0;0;0/ is a subspace of the full vector space R3.

What is a subspace of a vector space?

DEFINITIONA subspace of a vector space is a set of vectors (including 0) that satisfies two requirements: If v and w are vectors in the subspace and c is any scalar, then (i) v Cw is in the subspace and (ii) cv is in the subspace. In other words, the set of vectors is “closed” under addition v Cw and multiplication cv (and dw).

Do all vector spaces have to obey the 8 rules?

All vector spaces have to obey the eight reasonable rules. A real vector space is a set of “vectors” together with rules for vector addition and multiplication by real numbers. The addition and the multiplication must produce vectors that are in the space.