 # Question: Is R Infinity A Vector Space?

## Is C C a vector space?

Thus, C is a two-dimensional R-vector space (and, as any field, one-dimensional as a vector space over itself, C)..

## Is polynomial a vector space?

The set of all polynomials with real coefficients is a real vector space, with the usual oper- ations of addition of polynomials and multiplication of polynomials by scalars (in which all coefficients of the polynomial are multiplied by the same real number).

## Is CA vector space over Q?

This shows that C is a free Q-module, since that concept is literally equivalent to the existence of a basis, and the same argument in fact proves that any vector space is a free module over its field.

## What is not a vector space?

The following sets and associated operations are not vector spaces: (1) The set of n×n magic squares (with real entries) whose row, column, and two diagonal sums equal s≠0, with the usual matrix addition and scalar multiplication; (2) the set of all elements u of R3 such that ||u||=1, where ||⋅|| denotes the usual …

## Why r/c is not a vector space?

For example, R is not a vector space over C, because multiplication of a real number and a complex number is not necessarily a real number.

## Is R over QA vector space?

You need to show any basis of R over Q is infinite. In fact such a basis must be uncountably infinite. We can encode the structure of a vector space as a quadruple (V,K,s,m), where s:V×V→V and m:K×V→V satisfy the hypothesis in the axioms.

## What is R infinity?

We define R∞ as the set of infinite sequences (x1,x2,x3,…)

## Is a vector infinite?

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.

## Can 0 be a polynomial?

Like any constant value, the value 0 can be considered as a (constant) polynomial, called the zero polynomial. It has no nonzero terms, and so, strictly speaking, it has no degree either.

## Is r3 a vector space?

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.

## Is a line a vector space?

Since the set of lines in satisfies all ten vector space axioms under the defined operations of addition and multiplication, we have that thus is a vector space.

## Can a vector space have no basis?

Every vector space has a basis. Although it may seem doubtful after looking at the examples above, it is indeed true that every vector space has a basis. Let us try to prove this. First, consider any linearly independent subset of a vector space V , for example, a set consisting of a single non-zero vector will do.

## What is a high dimensional vector?

It may be easier to think of a high dimensional vector as simply describing quantities of distinct objects. For example, a five-dimensional vector could describe the numbers of apples, oranges, banana, pears, and cherries on the table. High-dimensional vectors have a lot of practical use.

## Is Hilbert space infinite dimensional?

Of all the infinite-dimensional topological vector spaces, the Hilbert spaces are the most “well-behaved” and the closest to the finite-dimensional spaces. One goal of Fourier analysis is to write a given function as a (possibly infinite) sum of multiples of given base functions.

## Is 0 a vector space?

The simplest example of a vector space is the trivial one: {0}, which contains only the zero vector (see the third axiom in the Vector space article). Both vector addition and scalar multiplication are trivial.