Table of Contents

## Can a vector space have only one element?

Any set with exactly one element defines a vector space over any field you like. Any set with exactly one element defines a vector space over any field you like. The axioms of a vector space require that there be a zero vector—since we only have one candidate, it must be that our single element is the zero vector.

**Can a vector space have one vector?**

In particular, a vector space with only the zero vector contains exactly one vector. No vector space can have two separate zero-vectors, as a consequence of the definition. It therefore follows that 0=0′. So, the two elements cannot be distinct.

**Which one is example for 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. A basis for this vector space is the empty set, so that {0} is the 0-dimensional vector space over F.

### What is an element of a vector space?

A vector space is a space in which the elements are sets of numbers themselves. Each element in a vector space is a list of objects that has a specific length, which we call vectors. We usually refer to the elements of a vector space as n-tuples, with n as the specific length of each of the elements in the set.

**Which one is not vector space?**

Most sets of n-vectors are not vector spaces. P:={(ab)|a,b≥0} is not a vector space because the set fails (⋅i) since (11)∈P but −2(11)=(−2−2)∉P. Sets of functions other than those of the form ℜS should be carefully checked for compliance with the definition of a vector space.

**Can a vector space have more than one zero vector?**

Every vector space contains a zero vector. True. The existence of 0 is a requirement in the definition. Thus there can be only one vector with the properties of a zero vector.

#### Is R4 a vector space?

A vector in n−space is represented by an ordered n−tuple (x1,x2,…,xn). 4. R4 = 4 − space = set of all ordered quadruples (x1,x2,x3,x4) of real numbers. A vector in the n−space Rn is denoted by (and determined) by an n−tuples (x1,x2,…,xn) of real numbers and same for a point in n−space Rn.

**What is the deﬁnition of vector space?**

The deﬁnition of a vector space is the same for F being R or C. A vector space V is a set of vectors with an operation of addition (+) that assigns an element u + v ∈ V to each u,v ∈ V. This means that V is closed under addition.

**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.

## 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.

**Does every vector space have a unique additive identity?**

Proposition 1. Every vector space has a unique additive identity. Proof. Suppose there are two additive identities 0 and 0′. Then 0 ′= 0+0 = 0, where the ﬁrst equality holds since 0 is an identity and the second equality holds since 0′ is an identity. Hence 0 = 0′ proving that the additive identity is unique. Proposition 2.