Can a vector space have only one element?

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.

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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.

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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?

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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.