Definition. A real vector space is a nonempty set \(V\) of objects, called vectors, with two operations, viz, addition and scalar multiplication, that satisfy the following properties for all vectors \(\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}\) in \(V\) and all scalars (real numbers) \(c\) and \(d\).

1. \(\overrightarrow{u}+\overrightarrow{v}\) is in \(V\).


2. \(\overrightarrow{u}+\overrightarrow{v}=\overrightarrow{v}+\overrightarrow{u}\)


3. \((\overrightarrow{u}+\overrightarrow{v})+\overrightarrow{w}=\overrightarrow{u}+(\overrightarrow{v}+\overrightarrow{w})\)


4. There is a zero vector \(\overrightarrow{0}\) such that \(\overrightarrow{u}+\overrightarrow{0}=\overrightarrow{u}\).


5. There is a vector \(-\overrightarrow{u}\) such that \(\overrightarrow{u}+(-\overrightarrow{u})=\overrightarrow{0}\).


6. \(c\overrightarrow{u}\) is in \(V\).


7. \(c(\overrightarrow{u}+\overrightarrow{v})=c\overrightarrow{u}+c\overrightarrow{v}\)


8. \((c+d)\overrightarrow{u}=c\overrightarrow{u}+d\overrightarrow{u}\).


9. \(c(d\overrightarrow{u})=(cd)\overrightarrow{u}\).


10. \(1\overrightarrow{u}=\overrightarrow{u}\).

Remark.

1. Scalars are elements of a field such as the set of real numbers and the set of complex numbers. If scalars are complex numbers, then \(V\) is called a complex vector space.


2. From the definition we have the following:
   1. \(0\overrightarrow{u}=\overrightarrow{0}\)
   
   
   2. \(c\overrightarrow{0}=\overrightarrow{0}\)
   
   
   3. \(-\overrightarrow{u}=(-1)\overrightarrow{u}\)
   
   

Example. The following are real vector spaces.

1. \(V_n\), the set of all vectors (directed line segments) in \(\mathbb R^n\).
   
   
   • Addition: Usual vector addition by the triangle/parallelogram law.
   
   
   • Scalar multiplication: Usual scalar multiplication of vectors.
   
   
2. \(\mathbb R^n\) and \(\mathbb C^n\).
   
   
   • Addition: Entrywise addition.
   
   
   • Scalar multiplication: Entrywise scalar multiplication.
   
   
3. \(\mathbb R^{\infty}\), the set of all real sequences \((a_n)=(a_1,a_2,a_3,\ldots)\).
   
   
   • Addition: Entrywise addition.
   
   
   • Scalar multiplication: Entrywise scalar multiplication.
   
   
4. \(P_n\), the set of all real polynomials of degree at most \(n\).
   
   
   • Addition: If \(\overrightarrow{p}(t)=a_0+a_1t+\cdots+a_nt^n\) and \(\overrightarrow{q}(t)=b_0+b_1t+\cdots+b_nt^n\), then \[(\overrightarrow{p}+\overrightarrow{q})(t)=(a_0+b_0)+(a_1+b_1)t+\cdots+(a_n+b_n)t^n.\]
   • Scalar multiplication: If \(\overrightarrow{p}(t)=a_0+a_1t+\cdots+a_nt^n\) and \(c\in \mathbb R\), then \[(c\overrightarrow{p})(t)=ca_0+ca_1t+\cdots+ca_nt^n.\]
5. \(F\), the set of all real-valued functions on a set \(D\).
   
   
   • Addition: \((\overrightarrow{p}+\overrightarrow{q})(x)=\overrightarrow{p}(x)+\overrightarrow{q}(x)\) for all \(\overrightarrow{p}, \overrightarrow{q}\in F\).
   
   
   • Scalar multiplication: \((c\overrightarrow{p})(x)=c\overrightarrow{p}(x)\) for all \(\overrightarrow{p}\in F\) and \(c\in \mathbb R\).
   
   
6. \(L(V,W)\), the set of all linear transformations \(T:V\to W\) where \(V\) and \(W\) are real vector spaces.
   
   
   • Addition: \((\overrightarrow{T}+\overrightarrow{S})(\overrightarrow{v})=\overrightarrow{T}(\overrightarrow{v}) +\overrightarrow{S}(\overrightarrow{v})\) for all \(\overrightarrow{v}\in V\).
   
   
   • Scalar multiplication: \((c\overrightarrow{T})(\overrightarrow{v})=c\overrightarrow{T}(\overrightarrow{v})\) for all \(\overrightarrow{v}\in V\) and \(c\in \mathbb R\).
   
   
7. \(M_{m,n}(\mathbb R)\), the set of all \(m\times n\) real matrices.
   
   
   • Addition: Entrywise addition.
   
   
   • Scalar multiplication: Entrywise scalar multiplication.
   
   

Last edited