5.1 Definition and Examples of Vector Spaces

Definition 5.1.1. A non-empty set \(\mathbb {F}\) is called a field if two binary operations \(+\) and \(.\) are defined on \(\mathbb {F}\), such that the following axioms are satisfied for all \(x,y,z\in \mathbb {F}\):

1.: \((x+y)+z=x+(y+z)\)
2.: \(x+y=y+x\)
3.: There is an element \(0\in \mathbb {F}\) such that \(x+0=x\).
4.: There is an element \(-x\in \mathbb {F}\) such that \(x+(-x)=0\).
5.: \((xy)z=x(yz)\)
6.: \(xy=yx\)
7.: There is an element \(1\in \mathbb {F}\) such that \(1.x=x\)
8.: There is an element \(x^{-1}\in \mathbb {F}\) such that \(x.x^{-1}=1\)
9.: \(x(y+z)=xy+xz\)

Example 5.1.2.

1.: The set \(\mathbb {R}\) of real numbers under ordinary addition and multiplication is a field.
2.: The set \(\mathbb {C}\) of complex numbers under ordinary addition and multiplication is a field.

Definition 5.1.3. A non-empty set \(V\) is called a vector space over a field \(\mathbb {F}\) if two operations, addition and scalar multiplication are defined on \(V\) such that the following hold for all \(u,v,w\in V\) and for all \(\alpha , \beta \in \mathbb {F}\),

i.: \(u+v\) is in \(V\).
ii.: \((u+v)+w=u+(v+w)\)
iii.: \(u+v=v+u\)
iv.: There is an element \(0\in V\) such that \(u+0=u\).
v.: There is an element \(-u\in V\) such that \(u+(-u)=0\)
vi.: \(\alpha u\) is in \(V\)
vii.: \(\alpha (u+v)=\alpha u+\alpha v\)
viii.: \((\alpha +\beta ) u=\alpha u+\beta u\)
xi.: \(1.u=u\)

The scalars will be real or complex numbers. The elements of a vector space are called vectors.

A vector space whose scalar field is the set of real numbers is called a real vector space. A complex vector space is a vector space with the set of complex numbers as the scalar field.

Example 5.1.4.

1.: The set \(\mathbb {R}\) of real numbers with the usual operations of addition and multiplication is a real vector space.
2.: The set \(\mathbb {C}\) of complex numbers with the usual operations of addition and multiplication is a complex vector space.
3.: The set \(\mathbb {R}^n\) of ordered \(n\) tuples of real numbers is a real vector space under component-wise addition and scalar multiplication.
4.: The set \(M_{mn}(\mathbb {R})\) of real matrices with the usual addition and scalar multiplication of matrices is a real vector space.
5.: The set \(C[a,b]\) of all continuous real valued functions of the closed, bounded interval \([a,b]\) is a vector space under the operations \((f+g)(x)=f(x)+g(x)\) and \((\alpha f)(x)=\alpha f(x)\) for all \(x\in [a,b]\).

The following are some of the basic properties of vector spaces.

Proposition 5.1.5. Let \(V\) be a vector space over a field \(\mathbb {F}\). Then:

1.: The zero element \(0\) is unique in \(V\).
2.: If \(u\in V\) then the element \(-u\in V\) such that \(u+(-u)=0\) is unique.
3.: If \(u,v,w\in V\) and \(u+v=u+w\) then \(v=w\).
4.: \(0\cdot u=0\) for all \(u\in V\).
5.: \(-(\alpha u)=(-\alpha u)=\alpha (-u)\) for all \(\alpha \in \mathbb {F}\), \(u\in V\).

Proof.

1.: Let \(0\) and \(0'\) in \(V\) be such that \(u+0=u\) and \(u+0'=u\) for all \(u\in V\). Let \(u=0\). Then \(0+0'=0=0'\)
2.: Let \(-u\) and \(u'\) in \(V\) be such that \(u+(-u)=0\) and \(u+u'=0\). Then \(u'=u'+0=u'+(u+(-u))=(u'+u)+(-u)=0+(-u)=-u\).
3.: If \(u+v=u+w\), then \((-u)+u+v=(-u)+u+w\). This means \(((-u)+u)+v=((-u)+u)+w\), so that \(0+v=0+w\), i.e. \(v=w\).
4.: \(0\cdot u=(0+0)\cdot u=0\cdot u+0\cdot u=0+0\cdot u+0\cdot u\) from (2) it follows that \(0\cdot u=0\)
5.: Let \(\alpha \in \mathbb {F}\) and \(u\in V\). Then \((\alpha u)+(-\alpha u)=0\) and \(u+(-u)=0\) imply \(\alpha u+\alpha (-u)=\alpha (u+(-u))=\alpha \cdot 0=0\). From \(\alpha u+(-(\alpha u))=0,\) \(\alpha u+\alpha (-u)=0\) and (2), it follows that \(-\alpha u=\alpha (-u).\) Also, \(\alpha +(-\alpha )=0\), so that \((\alpha +(-\alpha ))u=0.u=0\). Thus \(\alpha u +(-\alpha u)=0=\alpha u+(-(\alpha u))\), so that \((-\alpha )u=-(\alpha u)\). Hence \(-(\alpha u)=(-\alpha )(u)=\alpha (-u)\).

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