Section 4.1 Introduction
Recall the eight properties (
Proposition 2.1.1) of operation addition and scalar multiplication in
\(\R^n\text{.}\) Any non-empty set with two operations, addition and scalar multiplication satisfying the eight properties is known as vector space. More precisely we have the following definition.
Definition 4.1.1. Abstract Vector Space.
Let \(V\) be a nonempty set with two operations \(+\colon V\times V\to V\) defined by \(+(x,y):=x+y\) and multiplication \(\cdot \colon \R\times V\to V\) defined by \(\cdot(\alpha,x):=\alpha \cdot x\text{.}\) Satisfying the following properties:
A1: for all \(x,y\in V\text{,}\) \(x+y=y+x\text{.}\)
A2: for all \(x,y,z\in V\text{,}\) \(x+(y+z)=(x+y)+z\text{.}\)
A3: There exists \(\overline{0}\in V\) such that for all \(x\in V\text{,}\) \(\overline{0}=x+\overline{0}\text{.}\) This \(\overline{0}\) is called an additive identity.
A4: for each \(x\in V\text{,}\) there is a vector \(x'\in V\text{,}\) such that \(x+x'=x'+x=\overline{0}\text{.}\) This \(x'\) is called an additive inverse of \(x\text{.}\)
S1: for all \(\alpha\in \R\) and \(x,y \in V\text{,}\) \(\alpha(x+y)=\alpha x+\alpha y\text{.}\)
S2: for all \(\alpha,\beta \in \R\) and \(x \in V\text{,}\) \((\alpha+\beta) x=\alpha x+\beta y\text{.}\)
S3: for all \(\alpha,\beta \in \R\) and \(x \in V\text{,}\) \((\alpha\beta) x=\alpha (\beta x)y=\beta(\alpha x)\text{.}\)
S4: for all \(x\in V\text{,}\) \(1\cdot x=x\text{.}\)
The set \(V\) with \('+'\) and \('\cdot'\) is called a vector space over \(\R\text{.}\) Elements of \(V\) are called vectors.
Example 4.1.2.
\(\R^n\) with usual addition and scalar multiplication defined in the Section is a vector space over \(\R\text{.}\)
Example 4.1.3.
The set \(M_{mn}(\R)\text{,}\) the set of all \(m\times n\) real matrices with usual matrix addition and scalar multiplication by a real number is a vector space over \(\R\text{.}\)
Example 4.1.4.
Fix a natural number \(n\text{.}\) The set \({\cal P}_n(R)\text{,}\) the set of all polynomials of degree less than equal \(n\text{,}\) with usual polynomial addition and scalar multiplication by a real number is a vector space over \(\R\text{.}\)
Example 4.1.5.
Let \(V\) be the set of real-valued functions defined on an interval \([a, b]\text{.}\) For all \(f\) and \(g\) in \(V\) and \(\alpha\in \R\text{,}\) define addition and scalar multiplication, respectively, by
\begin{equation*}
(f+g)(x):=f(x)+g(x) \text{ and } (\alpha f)(x):=\alpha f(x)\text{.}
\end{equation*}
\(V\) is a vector space over \(\R\text{.}\)
Example 4.1.6.
The set of complex numbers \(\mathbb{C}:=\{a+ib:a,b\in \R\}\text{,}\) where \(i^2=-1\text{,}\) with addition and multiplication defined as
\begin{equation*}
(a_1+ib_1)+(a_2+ib_2):=(a_1+a_2)+i(b_1+b_2), \alpha (a+ib)=(\alpha a)+i(\alpha b)\text{.}
\end{equation*}
The set \((\mathbb{C},+,\cdot)\) is a vector space over \(\R\text{.}\)
Example 4.1.7.
Let \(X\) be any nonempty set and define \({\cal F}(X,\R):=\{f\colon X\to \R\}\text{,}\) the set of all functions from \(X\) to \(\R\text{.}\) Define addition and scalar multiplication, respectively, by
\begin{equation*}
(f+g)(x):=f(x)+g(x) \text{ and } (\alpha f)(x):=\alpha f(x)\text{.}
\end{equation*}
Then \(({\cal F}(X,\R),+,\cdot)\) is a vector space over \(\R\text{.}\)
Example 4.1.8.
(i) The set of rational numbers \(\mathbb{Q}\) with usual addition and multiplication is a vector space over \(\mathbb{Q}\text{.}\) However, \(Q\) is not a vector space over \(\R\text{.}\)
(ii) \(\R\) is a vector space over \(\mathbb{Q}\text{.}\)
Example 4.1.9.
Let \(V\) be a vector space over \(\R\) and \(X\) is a nonempty set. Let \(\varphi\colon X\to V\) be a bijection. We define addition and scalar multiplication on \(X\) as follows:
\begin{gather*}
\text{ for } x_1,x_2\in X, x_1+x_2:=\varphi(x_1)+\varphi(x_2) \\
\text{ and for } \alpha \in \R, x\in X \alpha\cdot x=\alpha\cdot \varphi(x).
\end{gather*}
It is easy to check that \(X\) with above addition and scalar multiplication is a vector space over \(\R\text{.}\)
Example 4.1.10.
Let \(V=(0,\infty)\text{.}\) Define addition and scalar multiplication on \((0,\infty)\) as follows:
\begin{equation*}
x+y:=xy \text{ and } \alpha \cdot x:= x^\alpha\text{.}
\end{equation*}
Check that
\((0,\infty)\) under this addition and scalar multiplication is a vector space over
\(\R\text{.}\) Contrast this example with Example
Example 4.1.9.
Example 4.1.11.
Let \(L=\{(x,y):y-x=1\}=\{(t,1+t):t\in \R\}\text{.}\) Define addition and scalar multiplication on \(L\) by
\begin{equation*}
(t_1,1+t_1)+(t_2,1+t_2):=(t_1+t_2,1+t_1+t_2), \alpha (t,1+t):=(\alpha t,1+\alpha t)\text{.}
\end{equation*}
Check that
\(L\) under this addition and scalar multiplication is a vector space over
\(\R\text{.}\) Contrast this example with Example
Example 4.1.9.
Example 4.1.12.
Let \(V=\{\bigstar\}\) be a singleton set. Define addition and scalar multiplication by
\begin{equation*}
\bigstar +\bigstar:=\bigstar \text{ and } \alpha \cdot \bigstar:= \bigstar , \alpha \in \R\text{.}
\end{equation*}
Check that \(V\) is a vector space over \(\R\) under the addition and scalar multiplication defined above.
Example 4.1.13.
Let \(V=\R^2\text{.}\) Define addition and scalar multiplication on \(\R^2\) as
\begin{equation*}
(x_1,x_2)\oplus (y_1,y_2):=(x_1+x_2+1,y_1+y_2+1)
\end{equation*}
and
\begin{equation*}
\alpha \odot (x_1,x_2):= (\alpha x_1+c-1,\alpha x_2+c-1)\text{.}
\end{equation*}
Check that \(\R^2,\oplus,\odot)\) is a vector space over \(\R\text{.}\) Find the bijection \(\varphi\colon \R^2\to \R^2\) is used to covert \(\R^2\) into a vector space using these operations. Find additive identity and the additive inverse of \((x_1,x_2)\) in \(\R^2\) corresponding to \(\oplus\text{.}\)
Example 4.1.14.
Consider the unit circle \(S=\{(x_1,x_2):x_1^2+x_2^2=1\}=\{(\cos t,\sin t):t\in \R\}\text{.}\) Define the addition and scalar multiplications by
\begin{equation*}
(\cos t,\sin t)+(\cos s,\sin s):=(\cos (t+s),\sin (t+s))
\end{equation*}
and
\begin{equation*}
\alpha \cdot (\cos t,\sin t):= (\cos (\alpha t),\sin (\alpha t))\text{.}
\end{equation*}
Show that \(S\) is a vector space over \(\R\) with respect to the addition and scalar multiplication defined above. Find the additive identity and additive inverse.
Example 4.1.15.
Let \(GL_n(\R)\) denote the set of all \(n\times n\) non singular real matrices. Define
\begin{equation*}
A\oplus B:=AB, \quad \text{and }\alpha\odot A:=\alpha A
\end{equation*}
where \(AB\) is the usual matrix multiplication, and \(\alpha A\) is the usual scalar multiplication. Show that \(GL_n(\R)\) is a vector space over \(\R\text{.}\)
Next we list the some properties in a vector space \(V\) over \(\R\text{.}\) These properties are easy to prove. Readers are encouraged to prove these properties.
Theorem 4.1.16. Properties.
Let \(V\) be a vector space over \(\R\text{.}\) Then we have the following properties:
(i) Additive identity in \(V\) is unique.
(ii) Additive inverse in \(V\) is unique.
(iii) \(0\cdot v=0\) for any \(v\in V\text{.}\)
(iv) \((-1)\cdot v=-v\) for all \(v\in V\text{.}\)
(v) \(-(-v) =v\) for all \(v\in V\text{.}\)
(vi) If \(\alpha \cdot v=0\) then either \(\alpha=0\) or \(v=0\text{.}\)
(vii) If \(v+u=u\text{,}\) then \(v=0\text{,}\) called the right cancellation. Similarly, we have left cancellation.
(viii) If \(\alpha v=\beta v\) and \(v\neq 0\text{,}\) then \(\alpha=\beta\text{.}\)
(ix ) \(\alpha v=\alpha u\) and \(\alpha\neq 0\text{,}\) then \(u=v\text{.}\)
In view of these, properties, here onward we will write “the additive identity” and “the additive inverse”.
