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Chapter 4: Problem 40

Prove that in a given vector space \(V\), the zero vector is unique.

Short Answer

Expert verified
The zero vector in a given vector space is unique. This conclusion was reached by defining the zero vector, assuming the existence of another zero vector that satisfies the same condition as the zero vector, and then showing that this assumption leads to the conclusion that the zero vectors are actually the same, proving the uniqueness.

Step by step solution

01

Define Zero Vector

By definition, zero vector is an element in vector space \(V\), denoted as \(0\), that when added to any vector \(v\) in \(V\), yields \(v\). Mathematically, for any \(v \in V\), we have \(v + 0 = v\).
02

Assume Existence of Another Zero Vector

To prove uniqueness, let's assume there exists another zero vector, denoted as \(0'\), that satisfies the same condition for all \(v \in V\), i.e., \(v + 0' = v\).
03

Prove the Uniqueness of Zero Vector

From the vector space properties, we know that for any vector \(v\), \(v + 0 = v\). Now, by replacing \(v\) with \(0\) in the assumed zero vector equation \(v + 0' = v\), we get \(0 + 0'=> 0\). But since \(0' + 0 = 0\), we can conclude that \(0' = 0\). Hence, zero vector is unique in a given vector space.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vector Space
A vector space is a fundamental concept in linear algebra where vectors can be added together and multiplied by scalars. It forms the basis of many mathematical theories and applications. In a vector space, a set of vectors, along with a set of scalars, or field, is defined. The following two main operations characterize a vector space:
  • Vector Addition: The addition of any two vectors in the space yields another vector in the same space.
  • Scalar Multiplication: Each vector in the space can be multiplied by a scalar, resulting in another vector in the same space.
Understanding vector spaces is crucial as it allows solving problems related to systems of linear equations, transformations, and geometric interpretations.
Uniqueness
Uniqueness, in the context of a vector space, refers to the idea that certain elements or properties are one-of-a-kind within that space. Specifically, when we talk about the zero vector being unique, it means that there is only one element in the vector space that behaves as the zero vector.
To prove the uniqueness of the zero vector, assume there are two zero vectors, say \(0\) and \(0'\). Both satisfy the condition for a zero vector: \(v + 0 = v\) and \(v + 0' = v\) for any vector \(v\).
  • If both satisfy this, then substituting \(v = 0\) in the second equation, we have \(0 + 0' = 0\).
  • Similarly, substituting \(v = 0'\) in the first gives \(0' + 0 = 0'\).
Combining these, you find that \(0 = 0'\), showing that the zero vector must be unique.
Vector Addition
Vector addition is one of the primary operations in a vector space. It is not just a simple combination of two vectors but involves several key properties that define how this operation interacts within the space.
  • Commutativity: The order of adding vectors doesn't matter. This means for any vectors \(u\) and \(v\), \(u + v = v + u\).
  • Associativity: Changing the grouping of vectors in addition doesn't affect the result. Mathematically, for any vectors \(u\), \(v\), and \(w\), \((u + v) + w = u + (v + w)\).
  • Existence of a Zero Vector: As proven, there is a unique zero vector \(0\) such that adding it to any vector \(v\) gives back the vector \(v\), i.e., \(v + 0 = v\).
These properties ensure that vector addition conforms to intuitive and predictable rules, making them reliable for computations and theoretical applications.
Properties of Vector Space
A vector space is defined by a collection of properties that vectors within the space must adhere to. Beyond just the operations of addition and scalar multiplication, these properties dictate how vectors interact and behave within the space.
  • Closure under Addition and Scalar Multiplication: Adding two vectors or multiplying a vector by a scalar results in a vector that is still within the space.
  • Existence of Zero Vector: There is a vector, known as the zero vector, that when added to any vector in the space, leaves it unchanged.
  • Existence of Additive Inverses: For every vector \(v\), there exists a vector \(-v\), such that \(v + (-v) = 0\).
  • Distributive Properties: Scalar multiplication distributes over vector addition, both from the right and left, ensuring consistent interactions between vectors and scalars. Mathematically, \(a(u + v) = au + av\) and \((a + b)v = av + bv\).
These properties are pivotal as they ensure mathematical operations within vector spaces are consistent and follow logical rules. They provide a structure that supports further mathematical and scientific exploration.

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