Introduction

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.

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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:

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for all \(x,y\in V\text{,}\) \(x+y=y+x\text{.}\)

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for all \(x,y,z\in V\text{,}\) \(x+(y+z)=(x+y)+z\text{.}\)

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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.

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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 \(x\text{.}\)

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for all \(\alpha\in \R\) and \(x,y \in V\text{,}\) \(\alpha(x+y)=\alpha x+\alpha y\text{.}\)

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for all \(\alpha,\beta \in \R\) and \(x \in V\text{,}\) \((\alpha+\beta) x=\alpha x+\beta y\text{.}\)

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for all \(\alpha,\beta \in \R\) and \(x \in V\text{,}\) \((\alpha\beta) x=\alpha (\beta x)y=\beta(\alpha x)\text{.}\)

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for all \(x\in V\text{,}\) \(1\cdot x=x\text{.}\)

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The set \(V\) with \('+'\) and \('\cdot'\) is called a vector space over \(\R\text{.}\) Elements of \(V\) are called vectors.

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Remark 4.1.2.

In the above definition one can replace \(\R\) by any field \((F).\) For it can be \(\Q\text{,}\) the set of rational numbers, \(\mathbb{C}\text{,}\) the set of complex number or even finite fields.However, in this text unless mention, we shall assume \(V\) a vector space over.

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Example 4.1.3.

\(\R^n\) with usual addition and scalar multiplication defined in the Section is a vector space over \(\R\text{.}\)

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Example 4.1.4.

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 \(\R\text{.}\)

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Example 4.1.5.

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 \(\R\text{.}\)

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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), \end{equation*}

\begin{equation*} \alpha\cdot (a+ib):=(\alpha a)+i(\alpha b). \end{equation*}

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The set \((\mathbb{C},+,\cdot)\) is a vector space over \(\R\text{.}\)

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Example 4.1.7.

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 \(\R\text{.}\)

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We can generalize Example 4.1.7 as follows.

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Example 4.1.8.

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 \(\R\text{.}\)

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Example 4.1.9.

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 \(\R\text{.}\)
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\(\R\) is a vector space over \(\mathbb{Q}\text{.}\)
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Example 4.1.10.

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*}

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It is easy to check that \(X\) with above addition and scalar multiplication is a vector space over \(\R\text{.}\)

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Example 4.1.11.

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*}

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Check that \((0,\infty)\) under this addition and scalar multiplication is a vector space over \(\R\text{.}\) Contrast this example with Example Example 4.1.10.

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Example 4.1.12.

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*}

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Check that \(L\) under this addition and scalar multiplication is a vector space over \(\R\text{.}\) Contrast this example with Example Example 4.1.10.

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Example 4.1.13.

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*}

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Check that \(V\) is a vector space over \(\R\) under the addition and scalar multiplication defined above.

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Example 4.1.14.

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*}

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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 \(\oplus\text{.}\)

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Example 4.1.15.

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*}

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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.

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Example 4.1.16.

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 \(\R\text{.}\)

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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.

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Theorem 4.1.17. Properties.

Let \(V\) be a vector space over \(\R\text{.}\) Then we have the following properties:

Additive identity in \(V\) is unique.
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Additive inverse in \(V\) is unique.
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\(0\cdot v=0\) for any \(v\in V\text{.}\)
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\((-1)\cdot v=-v\) for all \(v\in V\text{.}\)
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\(-(-v) =v\) for all \(v\in V\text{.}\)
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If \(\alpha \cdot v=0\) then either \(\alpha=0\) or \(v=0\text{.}\)
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If \(v+u=u\text{,}\) then \(v=0\text{,}\) called the right cancellation. Similarly, we have left cancellation.
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If \(\alpha v=\beta v\) and \(v\neq 0\text{,}\) then \(\alpha=\beta\text{.}\)
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\(\alpha v=\alpha u\) and \(\alpha\neq 0\text{,}\) then \(u=v\text{.}\)
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In view of these, properties, here onward we will write “the additive identity” and “the additive inverse”.

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