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Linear Algebra: Companion notes

Abhay Soman

Section 2.2 Examples of vector spaces

In this section we list some examples of vector spaces. More examples will be discussed in the class. In most of the instances the checking is left to the reader.
A trivial abelian group \(V=\{0\}\) over a field \(F\) is a vector space. The scalar multiplication is given by \(\alpha\cdot 0=0\) for \(\alpha\in F\text{.}\) Note that zero-dimensional vector spaces over different fields are different!
Let \(F\) be a field, and let
\begin{equation*} F^n=\{(x_1,x_2,\ldots,x_n):x_i\in F\}. \end{equation*}
We consider \(F^n\) as an abelian group by componentwise addition. We make \(F^n\) as an \(F\) vector space by defining scalar multiplication as follows:
\begin{equation*} \alpha\cdot(x_1,x_2,\ldots,x_n)=(\alpha x_1,\alpha x_2,\ldots,\alpha x_n). \end{equation*}
Note that \(\alpha x_i\) is the multiplication in the field \(F\) while \(\alpha\cdot(x_1,x_2,\ldots,x_n)\) is a scalar multiplication. Check that with this scalar multiplication \(F^n\) is an \(F\) vector space.
For \(\alpha\in\R\) consider the map
\begin{equation*} \ell_\alpha\colon\R\to\R \end{equation*}
defined by \(\ell(x)=\alpha x\) for \(x\in\R\text{.}\) Consider
\begin{equation*} \End_{\R}(\R)=\{\ell_\alpha :\alpha\in\R\}. \end{equation*}
Addition of functions give additive structure on \(\End_{\R}(\R)\text{.}\) We may define scalar multiplication by
\begin{equation*} a\cdot\ell_\alpha=\ell_{a\alpha}\quad\quad\text{for}\quad a,\alpha\in\R. \end{equation*}
Let \(F\) be a field and consider \(M_{m\times n}(F)\) the set of \(m\times n\) matrices over \(F\text{.}\) The additive structure on \(M_{m\times n}(F)\) can be given by usual addition of matrices. We define scalar multiplication as follows:
\begin{equation*} \alpha\cdot\begin{pmatrix}a_{11}\amp a_{12}\amp\cdots\amp a_{1n}\\a_{21}\amp a_{22}\amp\cdots\amp a_{2n}\\\vdots\amp\vdots\amp\ddots\amp\vdots\\a_{m1}\amp a_{m2}\amp\cdots\amp a_{mn}\end{pmatrix}=\begin{pmatrix}\alpha a_{11}\amp \alpha a_{12}\amp\cdots\amp \alpha a_{1n}\\\alpha a_{21}\amp \alpha a_{22}\amp\cdots\amp \alpha a_{2n}\\\vdots\amp\vdots\amp\ddots\amp\vdots\\\alpha a_{m1}\amp \alpha a_{m2}\amp\cdots\amp \alpha a_{mn}\end{pmatrix}\quad\text{for}\quad\alpha,a_{ij}\in F. \end{equation*}
Let \(R\) be a ring with unity \(1\text{,}\) and let \(F\) be a subfield in \(R\) (we always assume that the unity of a ring and its subring is the same). Then \(R\) is a vector space over \(F\) with the ring multiplication in \(R\) considered as a scalar multiplication. Note that elements of \(F\) can be considered as scalars as well as vectors.
Let \(V,W\) be vector spaces over a field \(F\text{.}\) Consider
\begin{equation*} V\oplus W=\{(v,w):v\in V\text{ and }w\in W\}. \end{equation*}
Additive structure is defined by
\begin{equation*} (v_1,w_1)+(v_2,w_2)=(v_1+v_2,w_1+w_2), \end{equation*}
where \(v_i\in V\) and \(w_i\in W\text{.}\) The addition \(v_1+v_2\) (resp., \(w_1+w_2\)) is defined by using additive structure in \(V\) (resp., \(W\)). We define scalar multiplication as follows:
\begin{equation*} \alpha\cdot(v,w)=(\alpha v,\alpha w)\quad\text{for}\quad\alpha\in F\text{ and }v\in V\text{ and }w\in W. \end{equation*}
Here, \(\alpha v\) (resp., \(\alpha w\)) is the scalar multiplication defined for \(V\) (resp., \(W\)). Check that \(V\oplus W\) is a vector space over \(F\text{.}\)
Let \(I\) be an indexing set and let \(V_i\) (\(i\in I\)) be vector spaces over a field \(F\text{.}\) Consider the (Cartesian) product \(\prod_{i\in I}V_i\text{,}\) this is the set of all functions \(f\colon I\to \bigcup_{i\in I}V_i\) such that \(f(i)\in V_i\text{.}\) This is a vector space over \(F\) with following operations: if \((\ldots,x_i,\ldots)\) and \((\ldots,y_i,\ldots)\) are any two elements of \(\prod_{i\in I} V_i\) then
  1. \((\ldots,x_i,\ldots)+(\ldots,y_i,\ldots)=(\ldots, x_i+y_i,\ldots)\text{;}\) component wise addition.
  2. For any \(\alpha\in F\) the scalar multiplication is define by
    \begin{equation*} \alpha\cdot(\ldots,x_i,\ldots)=(\ldots,\alpha x_i,\ldots). \end{equation*}
Consider the multiplicative group of rationals, \(\Q^\times\) and its subgroup \((\Q^\times)^2=\{r^2:r\in\Q^\times\}\text{.}\) Consider the quotient group \(V=\Q^\times/(\Q^\times)^2\text{.}\) Note that the group operation on \(V\) is multiplication, and the identity element of \(V\) is \(1(\Q^\times)^2\text{.}\) We claim that \(V\) is a vector space over \(\mathbb{F}_2\text{.}\) We define the scalar multiplication as follows.
\begin{equation*} \mathbb{F}_2\times V\to V \end{equation*}
given by
\begin{equation*} \big(0,r(\Q^\times)^2\big)\mapsto 1(\Q^\times)^2\quad\text{and}\quad \big(1,r(\Q^\times)^2\big)\mapsto r(\Q^\times)^2. \end{equation*}
The scalar multiplication by \(1\in\mathbb{F}_2\) is decided by one of the axioms of vector space, viz., \(1\cdot v=v\text{.}\) The scalar multiplication \(0\in\mathbb{F}_2\) is a consequence of the vector space axioms: \(0\cdot v=0\in V\) (\(0\in V\) is \(1\left(\Q^\times\right)\)).
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