We generalize the theory of vectors in [math]\mathbb{R}^n[/math] by extracting all the essential features of vector addition and scaling that make the theory works. The following is the most fundamental definition in linear algebra, which embodies all the important characteristics of vectors. [u][br][br]Definition[/u]: A [b]vector space over[/b] [math]\mathbb{R}[/math] is a nonempty set [math]V[/math] of objects, called [b]vectors[/b], on which are defined two operations, called [b]addition[/b] and [b]multiplication by scalar (real number)[/b], that satisfy the following [b]axioms[/b]: For any [math]u,v,[/math] and [math]w[/math] in [math]V[/math] and real numbers [math]c[/math] and [math]d[/math]:[br][list=1][*][math]u+v[/math] and [math]cu[/math] are in [math]V[/math] (Closed under addition and scalar multiplication)[/*][*][math]u+v=v+u[/math] (commutativity of addition)[/*][*][math]\left(u+v\right)+w=u+\left(v+w\right)[/math] (associativity of addition)[/*][*]There exists a [b]zero vector[/b] [math]0[/math] in [math]V[/math] such that [math]u+0=u[/math] (additive identity)[/*][*]For each [math]u[/math] in [math]V[/math], there exists a vector [math]-u[/math] in [math]V[/math] such that [math]u+\left(-u\right)=0[/math] (additive inverse)[/*][*][math]c\left(u+v\right)=cu+cv[/math] (distributivity for vector addition)[/*][*][math]\left(c+d\right)u=cu+du[/math] (distributivity for scalar addition)[/*][*][math]c\left(du\right)=\left(cd\right)u[/math] (Compatibility of scalar multiplication with real number multiplication)[br][/*][*][math]1u=u[/math]  (Identity of scalar multiplication)[/*][/list][br][u]Remark[/u]: More generally, we can consider scalars other than [math]\mathbb{R}[/math] in the above definition e.g. a vector space over [math]\mathbb{C}[/math].[br][br]We can deduce the following simple facts directly from the above axioms:[br][list][*]Zero vector [math]0[/math] is unique.[/*][*]For any vector [math]u[/math], [math]-u[/math] is unique. Moreover, [math]-u=\left(-1\right)u[/math].[/*][*][math]0u=0[/math] for any vector [math]u[/math]. (Note: the left zero is a real number and the right zero is the zero vector.)[/*][*][math]c0=0[/math] for any real number [math]c[/math]. [br][/*][/list]

The following are some examples of vector spaces over [math]\mathbb{R}[/math]:[br][br][br][u]Example 1[/u]: The first obvious example is [math]\mathbb{R}^n[/math].[br][br][br][u]Example 2[/u]: Polynomials[br][br]For non-negative integer [math]n[/math], [math]\mathbb{P}_n=[/math] the set of polynomials in [math]t[/math] of degree at most [math]n[/math] with real coefficients. Let [math]p(t)=a_0+a_1t+\cdots+a_nt^n[/math] and [math]q(t)=b_0+b_1t+\cdots+b_nt^n[/math] be polynomials in [math]\mathbb{P}_n[/math]. Then we define the following:[br][br][b]Addition[/b]: [math]p(t)+q(t)=(a_0+b_0)+(a_1+b_1)t+\cdots+(a_n+b_n)t^n[/math][br][br][b]Scalar multiplication[/b]: [math]cp(t)=ca_0+ca_1t+\cdots+ca_nt^n[/math] for any real number [math]c[/math][br][br]We consider such polynomials as "vectors". It can easily be shown that they satisfy all the axioms in the definition. Zero polynomial acts as the zero vector. Therefore, [math]\mathbb{P}_n[/math] is a vector space.[br][br][br][u]Example 3[/u]: Sequences of real numbers [br][br]Let [math]\mathbb{S}[/math] be the set of all real number sequences. We denote a real number sequence [math]a_1, a_2, a_3, \ldots [/math] by [math](a_n)[/math]. For any sequences [math](a_n)[/math] and [math](b_n)[/math] in [math]\mathbb{S}[/math], we define the following:[br][br][b]Addition[/b]: [math](a_n)+(b_n)=(a_n+b_n)[/math][br][br][b]Scalar multiplication[/b]: [math]c(a_n)=(ca_n)[/math] for any real number [math]c[/math][br][br]Again, it is easy to verify that [math]\mathbb{S}[/math] is a vector space.[br][br][br][u]Example 4[/u]: Matrices[br][br]Let [math]M_{m\times n}[/math] be the set of all m x n matrices. And we have already defined the addition and scalar multiplication of m x n matrices before. It is clear that [math]M_{m\times n}[/math] is a vector space. Hence, the set of all linear transformation from [math]\mathbb{R}^n[/math] to [math]\mathbb{R}^m[/math] is also a vector space.[br][br][br][u]Example 5[/u]: Real-valued functions[br][br]Let [math]V[/math] be the set of all real-valued functions defined on a set [math]D[/math]. Let [math]f:D\to \mathbb{R}[/math] and [math]g:D\to \mathbb{R}[/math] be two such functions in [math]V[/math]. Then we define the following: [br][br][b]Addition[/b]: [math](f+g)(x)=f(x)+g(x)[/math] for any [math]x[/math] in [math]D[/math][br][br][b]Scalar multiplication[/b]: Let [math]c[/math] be any real number. [math](cf)(x)=cf(x)[/math] for any [math]x[/math] in [math]D[/math][br][br]The real-valued functions can be regarded as "vectors" and it can be shown that they satisfy all the axioms in the definition of vector spaces. Therefore, [math]V[/math] is a vector space.[br][br][br][br]