Product and Quotient Spaces

Product Spaces

Definition 52: Product Space

Let $𝑉_{1}, 𝑉_{2}, \dots, 𝑉_{𝑛}$ be vector spaces over a field $𝐹$ . The product space (or just product) of $𝑉_{1}, 𝑉_{2}, \dots, 𝑉_{𝑛}$ , denoted $𝑉_{1} \times 𝑉_{2} \times \dots \times 𝑉_{𝑛}$ , is the vector space extended from the cartesian product $𝑉_{1} \times 𝑉_{2} \times \dots \times 𝑉_{𝑛}$ :

𝑉_{1} \times 𝑉_{2} \times \dots \times 𝑉_{𝑛} = {(𝐯_{1}, 𝐯_{2}, \dots, 𝐯_{𝑛}) | 𝐯_{1} \in 𝑉_{1}, \dots, 𝐯_{𝑛} \in 𝑉_{𝑛}},

with vector addition and scalar multiplication defined componentwise:

(𝐯_{1}, \dots, 𝐯_{𝑛}) + (𝐰_{1}, \dots, 𝐰_{𝑛}) = (𝐯_{1} + 𝐰_{1}, \dots, 𝐯_{𝑛} + 𝐰_{𝑛}), 𝑎 (𝐯_{1}, \dots, 𝐯_{𝑛}) = (𝑎 𝐯_{1}, \dots, 𝑎 𝐯_{𝑛}) .

That is, the product space $𝑉_{1} \times \dots \times 𝑉_{𝑛}$ is the cartesian product $𝑉_{1} \times \dots \times 𝑉_{𝑛}$ equipped with componentwise vector addition and scalar multiplication.

Theorem 48: Product Space is a Vector Space

Let $𝑉_{1}, \dots, 𝑉_{𝑛}$ be vector spaces over $𝐹$ . Then the product space $𝑉_{1} \times \dots \times 𝑉_{𝑛}$ is a vector space over $𝐹$ .

Notation

For a vector space $𝑉$ , the product space $𝑉^{𝑛}$ is defined as:

𝑉^{𝑛} = \underset{𝑛 times}{\underset{⏟}{𝑉 \times \dots \times 𝑉}} .

Example.

𝐹^{𝑛}

is the product space

𝐹 \times \dots \times 𝐹

Theorem 49: Dimension of a Product Space

Let $𝑉_{1}, \dots, 𝑉_{𝑛}$ be finite-dimensional vector spaces over $𝐹$ . Then the product space $𝑉_{1} \times \dots \times 𝑉_{𝑛}$ is also finite-dimensional, with:

\dim (𝑉_{1} \times \dots \times 𝑉_{𝑛}) = \dim (𝑉_{1}) + \dots + \dim (𝑉_{𝑛}) .

Corollary 29

For a finite-dimensional vector space $𝑉$ , $\dim (𝑉^{𝑛}) = 𝑛 \dim (𝑉)$ .

Definition 53: Summation Map

Let $𝑉_{1}, \dots, 𝑉_{𝑛}$ be vector spaces over a field $𝐹$ . The summation map $Γ : 𝑉_{1} \times \dots \times 𝑉_{𝑛} \to 𝑉_{1} + \dots + 𝑉_{𝑛}$ is the linear map defined by:

Γ (𝐯_{1}, \dots, 𝐯_{𝑛}) = 𝐯_{1} + \dots + 𝐯_{𝑛} .

Proposition 10

The summation map $Γ$ is indeed a linear map.

Theorem 50: Injectivity of the Summation Map

Let $𝑉_{1}, \dots, 𝑉_{𝑛}$ be vector spaces over a field $𝐹$ . Then $𝑉_{1} + \dots + 𝑉_{𝑛}$ is a direct sum if and only if the summation map $Γ : 𝑉_{1} \times \dots \times 𝑉_{𝑛} \to 𝑉_{1} + \dots + 𝑉_{𝑛}$ is injective

¹.

Quotient Spaces

Notation

Let $𝑉$ be a vector space, and suppose $𝐯 \in 𝑉$ . For any $𝑈 \subseteq 𝑉$ , the set $𝐯 + 𝑈$ is defined by:

𝐯 + 𝑈 = {𝐯 + 𝐮 | 𝐮 \in 𝑈} .

The set $𝐯 + 𝑈$ is called the translate of $𝑈$ by $𝐯$ .

Definition 54: Quotient Space

Let $𝑉$ be a vector space over a field $𝐹$ , and let $𝑈 \subseteq 𝑉$ be a subspace of $𝑉$ . The quotient space (or just quotient) of $𝑉$ by $𝑈$ , denoted $𝑉 / 𝑈$ , is the set of all translates of $𝑈$ by vectors in $𝑉$ :

𝑉 / 𝑈 = {𝐯 + 𝑈 | 𝐯 \in 𝑉} .

Definition 55: Quotient Map

Let $𝑉$ be a vector space over a field $𝐹$ , and let $𝑈 \subseteq 𝑉$ be a subspace of $𝑉$ . The quotient map $𝜋 : 𝑉 \to 𝑉 / 𝑈$ is the linear map defined by:

𝜋 (𝐯) = 𝐯 + 𝑈 .

Proposition 11

The quotient map $𝜋$ is indeed a linear map.

Theorem 51: Dimension of a Quotient Space

Let $𝑉$ be a finite-dimensional vector space over a field $𝐹$ , and let $𝑈 \subseteq 𝑉$ be a subspace of $𝑉$ . Then the quotient space $𝑉 / 𝑈$ is also finite-dimensional, with:

\dim (𝑉 / 𝑈) = \dim (𝑉) - \dim (𝑈) .

Definition 56: Induced Maps on Quotient Spaces

Suppose $𝑇 : 𝑉 \to 𝑊$ is a linear map. The induced map of $𝑇$ , denoted $\tilde{𝑇}$ , is the linear map $\tilde{𝑇} : 𝑉 / \ker (𝑇) \to 𝑊$ defined by:

\tilde{𝑇} (𝐯 + \ker (𝑇)) = 𝑇 (𝐯) .

Proposition 12

The induced map $\tilde{𝑇}$ is indeed a linear map.

Corollary 30

Let $𝜋$ be the quotient map of $𝑉$ onto $𝑉 / \ker (𝑇)$ . Then $\tilde{𝑇} \circ 𝜋 = 𝑇$ .

Exercises

Exercise 71.

For any vector space

𝑉

and

𝑛 \geq 1

, show that

𝑉^{𝑛}

is isomorphic to

ℒ (𝐹^{𝑛}, 𝑉)

¹ Since surjectivity of $Γ$ is trivial, we could also say $𝑉_{1} + \dots + 𝑉_{𝑛}$ is a direct sum iff $Γ$ is invertible.