Question

Prove that a vector space has only one zero element $\mathbf{0}$.

   Prove that a vector space has only one zero element $\mathbf{0}$.

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 2, Problem 12

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Step 1

In a vector space $V$ over a field $F$, a zero element (or zero vector) $\mathbf{0}$ is a unique element such that for any vector $\mathbf{v} \in V$, the addition of $\mathbf{v}$ and $\mathbf{0}$ results in $\mathbf{v}$ itself. That is, $\mathbf{v} + \mathbf{0} = Show more…

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Prove that a vector space has only one zero element $\mathbf{0}$.

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Key Concepts

Vector Space

A vector space is a mathematical structure consisting of a set of elements along with two operations—vector addition and scalar multiplication—that follow specific axioms such as associativity, commutativity, distributivity, and the existence of additive identities and inverses. This structure underlies much of linear algebra and its applications.

Additive Identity

The additive identity in a vector space is the unique element (often called the zero vector) that, when added to any vector in the space, leaves the vector unchanged. It is one of the defining axioms of a vector space and is crucial for the structure to behave like an abelian group under addition.

Proof of Uniqueness

The proof of the uniqueness of the additive identity relies on the general algebraic technique of showing that if two elements both satisfy the role of an identity, then they must be equal. This argument is rooted in the properties of the operations defined in the vector space and is a common method used in abstract algebra to establish the uniqueness of identity elements.

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