# Is the complex field a vector space?

## Is the complex field a vector space?

For example, the complex numbers C are a two-dimensional real vector space, generated by 1 and the imaginary unit i. The latter satisfies i2 + 1 = 0, an equation of degree two. Thus, C is a two-dimensional R-vector space (and, as any field, one-dimensional as a vector space over itself, C).

### Are polynomials a vector space over R?

The set of all polynomials are an (countably infinite dimensional) vector space. The set of continuous functions R → R are an (uncountably infinite dimensional) vector space.

#### Is the set of all polynomials of degree 3 a vector space?

It is stated that V, the set of all polynomials of degree exactly 3 is not a vector space. The reason the textbook gives is that this set does not contain a zero vector.

Is set of all polynomials 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 C 2 over Z vector space?

For example, the set C2 is also a real vector space under the same addition as before, but with multiplication only by real scalars, an operation we might denote ⋅R.

## Is C over R vector space?

(i) Yes, C is a vector space over R. Since every complex number is uniquely expressible in the form a + bi with a, b ∈ R we see that (1, i) is a basis for C over R. Thus the dimension is two. (ii) Every field is always a 1-dimensional vector space over itself.

### Is R NA vector space?

Definition and structures For any natural number n, the set Rn consists of all n-tuples of real numbers (R). With componentwise addition and scalar multiplication, it is a real vector space. Every n-dimensional real vector space is isomorphic to it.

#### Why are polynomials not a vector space?

Polynomials of degree n does not form a vector space because they don’t form a set closed under addition.

How do you prove vector space?

Prove Vector Space Properties Using Vector Space Axioms

1. Using the axiom of a vector space, prove the following properties.
2. (a) If u+v=u+w, then v=w.
3. (b) If v+u=w+u, then v=w.
4. (c) The zero vector 0 is unique.
5. (d) For each v∈V, the additive inverse −v is unique.
6. (e) 0v=0 for every v∈V, where 0∈R is the zero scalar.

Why r/c is not a vector space?

a vector space over its over field. 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. respect to the addition of matrices as vector addition and multiplication of a matrix by a scalar as scalar multiplication.

## Is every polynomial a vector?

Every polynomial has an equivalent finite-length vector. Every finite-length vector corresponds to a polynomial. What we are interested in, is the coefficients, not in the list of function values.

### What is polynomial space?

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#### What is polynomial basis?

In mathematics, a polynomial basis is a basis of a polynomial ring, viewed as a vector space over the field of coefficients, or as a free module over the ring of coefficients. The most common polynomial basis is the monomial basis consisting of all monomials.