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Chapter 30: Problem 26

Let \(V\) be a vector space over a field \(F\), and let \(S\) be a subspace of \(V\). Define the quotient space \(V / S\), and show that it is a vector space over \(F\).

Short Answer

Expert verified
The quotient space \(V / S\) is a vector space with cosets as elements, operations on cosets are well-defined, making \(V / S\) a vector space over \(F\).

Step by step solution

01

Defining the Quotient Space

The quotient space \(V / S\) is the set of equivalence classes of vectors in \(V\) under the equivalence relation defined by \(v \sim w\) if and only if \(v - w \in S\). Each equivalence class is called a coset, and represented by \(v + S\) for any vector \(v \in V\).
02

Coset Operations

Define the addition of two cosets \((v + S) + (w + S) = (v + w) + S\). Define scalar multiplication by \(\alpha(v + S) = (\alpha v) + S\) for a scalar \(\alpha \in F\). These operations need to be well-defined, meaning they do not depend on the choice of representative vectors from each coset.
03

Showing Well-Defined Operations: Addition

Suppose \(v_1 + S = v_2 + S\) and \(w_1 + S = w_2 + S\). Then \(v_1 - v_2 \in S\) and \(w_1 - w_2 \in S\). We want \((v_1 + w_1) + S = (v_2 + w_2) + S\). We know \((v_1 + w_1) - (v_2 + w_2) = (v_1 - v_2) + (w_1 - w_2) \), which belongs to \(S\), so addition is well-defined.
04

Showing Well-Defined Operations: Scalar Multiplication

For scalar multiplication, suppose \(v_1 + S = v_2 + S\). Then \(v_1 - v_2 \in S\). We want \(\alpha v_1 + S = \alpha v_2 + S\). Since \(\alpha(v_1 - v_2) = \alpha v_1 - \alpha v_2 \in S\) (because \(S\) is a subspace and closed under scalar multiplication), scalar multiplication is well-defined.
05

Vector Space Axioms

Verify that \(V / S\) satisfies vector space axioms: closure, associativity, commutativity, existence of a zero element, existence of additive inverses, distributive properties, and compatibility with field multiplication. All hold because operations are derived from those in \(V\) and \(S\) is a subspace (closed under these operations).

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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, comprising a collection of vectors that can be added together and multiplied by scalars. To be a vector space, the set along with these operations must satisfy a variety of axioms. These include:
  • Closure under addition and scalar multiplication: If you add two vectors or multiply a vector by a scalar, the result must still belong within the set.
  • Associativity and commutativity of addition: Vector addition must be consistent, regardless of the order in which vectors are added.
  • Existence of an additive identity and additive inverses: There must be a zero vector that can be added to any vector without changing it. Each vector must also have an opposite vector that sums to the zero vector.
  • Distributive and compatibility properties: These properties align vector addition and scalar multiplication with the underlying field's operations.
To help visualize, think of a vector space as a boundlessly expansive grid where each point (or vector) on the grid has defined rules for combining with other points, remaining within the grid's bounds.
Subspace
In linear algebra, a subspace is essentially a smaller vector space contained within a larger vector space. To be a subspace, a collection of vectors must itself satisfy all vector space axioms but relative to the larger vector space.
  • Closure: When you add two vectors from the subspace or scale a vector, it must remain within the subspace.
  • Contains the zero vector: The zero vector of the larger space must also be a member of any subspace.
  • Closed under linear combinations: Any linear combination of vectors from a subspace must also be within that subspace.
Subspaces are important because they help break down complex vector spaces into more manageable components. For instance, each plane or line through the origin in a three-dimensional space is a subspace.
Scalar Multiplication
Scalar multiplication is a key operation in a vector space. It involves multiplying a vector by a scalar, which is an element from the field over which the vector space is defined. This operation must satisfy specific conditions:
  • Closure: The result of the scalar multiplication must be a vector still within the vector space.
  • Distributive properties: Distributing a scalar over the addition of vectors, and distributing a sum of scalars over a single vector, should hold.
  • Associativity with field multiplication: Multiplying a vector by a scalar product should yield the same result as first scaling the vector by one of the scalars and then the other.
  • Identity element of multiplication: Multiplying a vector by the scalar one must return the original vector.
Think of scalar multiplication as stretching or compressing a vector. It's like pulling on a rubber band to change its length but not its direction, or squeezing it to shorten it without direction change.
Equivalence Relation
An equivalence relation is a way of grouping elements into classes based on certain criteria of similarity or equivalence. For vectors, an equivalence relation ties into the concept of cosets used in quotient spaces.
  • Reflexivity: Any vector is equivalent to itself.
  • Symmetry: If one vector is equivalent to another, then the second vector is equivalent to the first.
  • Transitivity: If a vector is equivalent to a second vector, and that second vector equivalent to a third, the first and third vectors are equivalent.
In the construction of a quotient space, vectors are deemed equivalent if their difference lies in a particular subspace. This equivalence allows us to form equivalence classes or cosets, which are the foundational elements of the quotient space, grouping vectors that are "similar" by the subspace's properties.

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Most popular questions from this chapter

\- Let \(F\) be amy ficld. Consider the "system of \(m\) simultancous lincar equations in \(n\) unknowns" $$ \begin{gathered} a_{11} X_{1}+a_{12} X_{2}+\cdots+a_{1 n} X_{n}=b_{1} \\ a_{21} X_{1}+a_{22} X_{2}+\cdots+a_{24} X_{n}=b_{2} \\ \vdots \\ a_{m i} X_{1}+a_{m 2} X_{2}+\cdots+a_{m n} X_{n}=b_{m} \end{gathered} $$ where \(a_{i j}, b_{i} \in F_{0}\) a. Show that the "system has a solution," that is, there exist \(X_{1}, \cdots, X_{n} \in F\) that satisfy all \(m\) equations, if and only if the vector \(\beta=\left(b_{1}, \cdots, b_{m}\right)\) of \(F^{m}\) is a linear combination of the vectors \(\alpha_{j}=\left(a_{1 j}, \cdots, a_{m j}\right) .\) (This result is straightforward to prove, being practically the definition of a solution, but should really be regarded as the fwndamental existence theorem for a simultancous solution of a system of linear equations.) b. From part (a), show that if \(n=m\) and \(\left\\{\alpha_{j} \mid j=1, \cdots, n\right]\) is a basis for \(F^{n}\), then the system always has a unique solution.

Give a basis for the indicated vector space over the field. $$ Q(\sqrt{2}) \text { over } Q $$

Correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication. The vectors in a subset \(S\) of a vector space \(V\) over a field \(F\) span \(V\) if and only if each \(\beta \in V\) can be expressed uniquely as a linear combination of the vectors in \(S\).

Find three bases for \(\mathbb{R}^{2}\) over \(\mathbb{R}\), no two of which have a vector in common.

The dimension over \(F\) of a finite-dimensional vector space \(V\) over a field \(F\) is the minimum number of vectors required to span \(V\).
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