Suppose that V and W are vector spaces. The Cartesian...

Suppose that $V$ and $W$ are vector spaces. The Cartesian product space, denoted by $V \times W$, is defined as the set of all ordered pairs $(\mathbf{v}, \mathbf{w})$, where $\mathbf{v} \in V, \mathbf{w} \in W$, with vector addition $(\mathbf{v}, \mathbf{w})+(\widehat{\mathbf{v}}, \widehat{\mathbf{w}})=(\mathbf{v}+\widehat{\mathbf{v}}, \mathbf{w}+\widehat{\mathbf{w}})$ and scalar multiplication $c(\mathbf{v}, \mathbf{w})=(c \mathbf{v}, c \mathbf{w})$. (a) Prove that $V \times W$ is a vector space. (b) Explain why $\mathbb{R} \times \mathbb{R}$ is the same as $\mathbb{R}^2$. (c) More generally, explain why $\mathbb{R}^m \times \mathbb{R}^n$ is the same as $\mathbb{R}^{m+n}$.

Suppose that $V$ and $W$ are vector spaces. The Cartesian product space, denoted by $V \times W$, is defined as the set of all ordered pairs $(\mathbf{v}, \mathbf{w})$, where $\mathbf{v} \in V, \mathbf{w} \in W$, with vector addition $(\mathbf{v}, \mathbf{w})+(\widehat{\mathbf{v}}, \widehat{\mathbf{w}})=(\mathbf{v}+\widehat{\mathbf{v}}, \mathbf{w}+\widehat{\mathbf{w}})$ and scalar multiplication $c(\mathbf{v}, \mathbf{w})=(c \mathbf{v}, c \mathbf{w})$.
(a) Prove that $V \times W$ is a vector space. (b) Explain why $\mathbb{R} \times \mathbb{R}$ is the same as $\mathbb{R}^2$.
(c) More generally, explain why $\mathbb{R}^m \times \mathbb{R}^n$ is the same as $\mathbb{R}^{m+n}$.

Key Concepts

Isomorphism and Dimensionality

In finite-dimensional linear algebra, two vector spaces are said to be isomorphic if there exists a one-to-one correspondence between them that preserves vector operations. The Cartesian product of ?^m and ?^n is isomorphic to ?^(m+n) because each element can be uniquely represented as an (m+n)-tuple. This reflects the principle that the dimension of the product space is the sum of the dimensions of the individual spaces, a key concept in understanding the structure and properties of finite-dimensional vector spaces.

Vector Space Operations on Product Spaces

Defining vector space operations on a product space involves applying the operations independently on each of the components. Specifically, the sum of two ordered pairs is defined as the ordered pair of their respective sums, and scalar multiplication is applied to each component. This component-wise definition is essential in proving that the product set satisfies the axioms of a vector space, ensuring its validity as a vector space.

Vector Space

A vector space is a fundamental structure in linear algebra consisting of a set equipped with two operations: vector addition and scalar multiplication. These operations satisfy several axioms such as associativity, commutativity, existence of an additive identity and inverses, and distributivity, among others. Understanding these axioms is crucial as they underpin the structure and behavior of the space regardless of the specific vectors or fields involved.

Cartesian Product of Sets

The Cartesian product is a construction that forms a new set from two sets by pairing every element of one set with every element of the other. In the context of vector spaces, when we form the Cartesian product of two vector spaces and define operations component-wise, the resulting set naturally inherits the structure of a vector space. This method of construction is foundational in creating higher-dimensional spaces from lower-dimensional ones.

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