Product and Quotient of Vector Spaces¶

All the vector spaces considered for a product or quotient must be over the same type of \(\field\) (\(\real\) or \(\comp\)).

Product of Vector Spaces¶

Suppose we have \(m\) vector spaces \(V_{1}, V_{2}, \ldots, V_{m} \in \field\), then their vector product

\[ \begin{aligned} V_{1}\times \cdots \times V_{m} = \{ (v_{1}, \ldots, v_{m}) \: v_{1}\in V_{1}, \ldots v_{m} \in V_{m} \}\end{aligned} \]

Addition and scalarm multiplication are also defined on such products

\[ \begin{aligned} (u_{1}, \ldots, u_{m}) + (v_{1}, \ldots, v_{m}) &= (u_{1} + v_{1}, \ldots, u_{m} + v_{m})\newline \lambda (v_{1}, \ldots, v_{m}) &= (\lambda v_{1}, \ldots, \lambda v_{m})\end{aligned} \]

where \(\lambda \in \field\). With the above definitions of addition and scalar multiplication, the product of vector spaces is itself a vector space.

\(\real^{2} \times \real^{3}\) is an example of a product of vector spaces. Elements of such a product are lists of the form \(((x_{1}, x_{2}), (x_{3}, x_{4}, x_{5}))\) where all 5 elements belong in \(\real\). Note that this product is technically different from \(\real^{5}\), but is a simple relabelling of \(\real^{5}\).

Dimension of a product of finite dimensional vector spaces is simply the sum of dimensions of the individual vector spaces

\[ \begin{aligned} \text{dim}(V_{1} \times \cdots \times V_{m}) = \text{dim }V_{1} + \cdots + \text{ dim }V_{m}\end{aligned} \]

and can be shown by considering the bases vectors on either side.

Quotient of Vector Spaces¶

We first define the meaning of adding a vector and a subspace. Suppose \(v \in \setv\) and \(U\) is a subspace of \(\setv\). Then

\[ \begin{aligned} v + U = \{ v + u \: u \in U \}\end{aligned} \]

For instance, consider \(U = \{(x, 3x) \in \real^{2} \: x \in \real \}\) and \(v = (2,3)\). Then \(v + U\) is the line with slope \(3\) containing the point \((2,3)\).

Affine Subset and Parallel

An affine subset of \(\setv\) is a subspace of \(\setv\) of the form \(v + U\) where \(v \in \setv\) and \(U\) is a subspace of \(\setv\). Such a subset is said to be parallel to \(U\).

Clearly, in the previous example, the subset \(\{(x + 2, 3x + 3) \: x \in \field \}\) is parallel to \(\{(x, 3x) \: x \in \field \}\) both by definition and physically (they are parallel lines).

Quotient Space \(\boldsymbol{\setv/U}\)

For any subspace \(U\) of \(\setv\), the quotient space \(\boldsymbol{\setv/U}\) is the set of all affine subsets of \(\setv\) parallel to \(U\).

\[ \begin{aligned} \setv/U &= \{ v + U \: v \in \setv \} \text{dim } \setv/U &= \text{ dim } \setv - \text{ dim } U\end{aligned} \]

Two affine subsets parrallel to \(U\) are equal or disjoint. Suppose \(v, w \in \setv\) and \(U\) is a subspace of \(\setv\), then the following three statements will be equivalent

\(v - w \in U\)
\(v + U = w + U\)
\((v + U) \cap (w + U) \neq \phi\)

Addition and Scalar Multiplication

Addition and scalar multiplication on \(\setv/U\) are defined as follows

\[ \begin{aligned} (v + U) + (w + U) &= (v + w) + U\newline \lambda(v + U) &= (\lambda v) + U\end{aligned} \]

With these definitions, \(\setv/U\) is also a vector space.

Quotient Map \(\boldsymbol{\pi}\) Suppose \(U\) is a subspace of \(\setv\), then the quotient map is a linear map from \(\setv\) to \(\setv/U\) defined as

\[ \begin{aligned} \pi = \{ v + U \: v \in \setv \}\end{aligned} \]