Quotient spaces: Let V be a vector space and W ⊂V a subspace. We say that two vectors 𝐮, 𝐯 ∈V ar

step 1 Since the dimension of S is n minus 1, there's exist a non -zero linear function. It takes input

Quotient spaces: Let V be a vector space and W ⊂V a...

Quotient spaces: Let $V$ be a vector space and $W \subset V$ a subspace. We say that two vectors $\mathbf{u}, \mathbf{v} \in V$ are equivalent modulo $W$ if $\mathbf{u}-\mathbf{v} \in W$. (a) Show that this defines an equivalence relation, written $\mathbf{u} \sim_W \mathbf{v}$ on $V$, i.e., $(i) \mathbf{v} \sim_W \mathbf{v}$ for every $\mathbf{v}$; (ii) if $\mathbf{u} \sim_W \mathbf{v}$, then $\mathbf{v} \sim_W \mathbf{u}$; and (iii) if, in addition, $\mathbf{v} \sim_W \mathbf{z}$, then $\mathbf{u} \sim_W \mathbf{z}$. (b) The equivalence class of a vector $\mathbf{u} \in V$ is defined as the set of all equivalent vectors, written $[\mathbf{u}]_W=\left\{\mathbf{v} \in V \mid \mathbf{v} \sim_W \mathbf{u}\right\}$. Show that $[\mathbf{0}]_W=W$. (c) Let $V=\mathbb{R}^2$ and $W=\left\{(x, y)^T \mid x=2 y\right\}$. Sketch a picture of several equivalence classes as subsets of $\mathbb{R}^2$. (d) Show that each equivalence class $[\mathbf{u}]_W$ for $\mathbf{u} \in V$ is an affine subspace of $V$, as in Exercise 2.2.28. (e) Prove that the set of equivalence classes, called the quotient space and denoted by $V / W=\{[\mathbf{u}] \mid \mathbf{u} \in V\}$, forms a vector space under the operations of addition, $[\mathbf{u}]_W+[\mathbf{v}]_W=[\mathbf{u}+\mathbf{v}]_W$, and scalar multiplication, $c[\mathbf{u}]_W=[c \mathbf{u}]_W$. What is the zero element? Thus, you first need to prove that these operations are well defined, and then demonstrate the vector space axioms.

Key Concepts

Vector Space Axioms

The vector space axioms are a collection of requirements—including associativity, commutativity, the existence of an additive identity and inverses, and distributivity of scalar multiplication—which must be satisfied for any structure to be considered a vector space. In constructing a quotient space, one must verify that the induced operations on equivalence classes adhere to these axioms, ensuring the structure maintains the essential characteristics of a vector space.

Affine Subspace

An affine subspace is a translation of a linear subspace; it can be thought of as a flat, shifted copy of a vector space that does not necessarily pass through the origin. Each equivalence class in a quotient space is an affine subspace of the original vector space, reflecting the idea that these classes are shifts of the subspace that defined the equivalence relation.

Quotient Space

A quotient space is the set of equivalence classes formed by a vector space when partitioned by an equivalence relation induced by a subspace. It inherits a vector space structure through the well-defined operations of addition and scalar multiplication on these equivalence classes, thereby allowing the study of the original space modulo the subspace.

Well-defined Operations

For the structure of a quotient space to be meaningful, the operations of addition and scalar multiplication on equivalence classes need to be well defined; that is, the result of these operations should not depend on the representatives chosen from the classes. Establishing this is crucial in proving that the set of equivalence classes indeed forms a vector space.

Equivalence Class

An equivalence class under a given equivalence relation is the set of all elements that are equivalent to a particular element. In quotient spaces, each equivalence class consists of all vectors that differ from a fixed vector by an element of a subspace, effectively grouping vectors that share a common 'offset' defined by the subspace.

Equivalence Relation

An equivalence relation is a binary relation that is reflexive, symmetric, and transitive. In the context of quotient spaces, the relation is defined such that two vectors are considered equivalent if their difference lies in a specified subspace. This foundational concept allows us to partition a vector space into disjoint equivalence classes.

Subspace

A subspace is a subset of a vector space that is itself a vector space under the same operations. In the study of quotient spaces, the subspace is used to determine the equivalence relation by providing the set of vectors whose inclusion as differences between elements defines equivalence.

Transcript

Please give Ace some feedback