Editing Vector space (section)

==Examples==
{{main|Examples of vector spaces}}

===Arrows in the plane===
<div class=skin-invert-image>{{multiple image
 | total_width = 200
 | direction = vertical
 | image1 = Vector addition3.svg
 | caption1 = Vector addition: the sum {{math|'''v''' + '''w'''}} (black) of the vectors {{math|'''v'''}} (blue) and {{math|'''w'''}} (red) is shown.
 | image2 = Scalar multiplication.svg
 | caption2 = Scalar multiplication: the multiples {{math|−'''v'''}} and {{math|2'''w'''}} are shown.
}}</div>
The first example of a vector space consists of [[arrow (symbol)|arrow]]s in a fixed [[plane (geometry)|plane]], starting at one fixed point. This is used in physics to describe [[force]]s or [[velocity|velocities]].{{sfn|Kreyszig|2020|p=[https://books.google.com/books?id=w4T3DwAAQBAJ&pg=PA355 355]}} Given any two such arrows, {{math|'''v'''}} and {{math|'''w'''}}, the [[parallelogram]] spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the ''sum'' of the two arrows, and is denoted {{math|'''v''' + '''w'''}}. In the special case of two arrows on the same line, their sum is the arrow on this line whose length is the sum or the difference of the lengths, depending on whether the arrows have the same direction. Another operation that can be done with arrows is scaling: given any positive [[real number]] {{math|''a''}}, the arrow that has the same direction as {{math|'''v'''}}, but is dilated or shrunk by multiplying its length by {{math|''a''}}, is called ''multiplication'' of {{math|'''v'''}} by {{math|''a''}}. It is denoted {{math|''a'''''v'''}}. When {{math|''a''}} is negative, {{math|''a'''''v'''}} is defined as the arrow pointing in the opposite direction instead.{{sfn|Kreyszig|2020|p=[https://books.google.com/books?id=w4T3DwAAQBAJ&pg=PA358 358&ndash;359]}}

The following shows a few examples: if {{math|1=''a'' = 2}}, the resulting vector {{math|''a'''''w'''}} has the same direction as {{math|'''w'''}}, but is stretched to the double length of {{math|'''w'''}} (the second image). Equivalently, {{math|2'''w'''}} is the sum {{math|'''w''' + '''w'''}}. Moreover, {{math|1=(−1)'''v''' = −'''v'''}} has the opposite direction and the same length as {{math|'''v'''}} (blue vector pointing down in the second image).

===Ordered pairs of numbers===
A second key example of a vector space is provided by pairs of real numbers {{mvar|x}} and {{mvar|y}}. The order of the components {{mvar|x}} and {{mvar|y}} is significant, so such a pair is also called an [[ordered pair]]. Such a pair is written as {{math|(''x'', ''y'')}}. The sum of two such pairs and the multiplication of a pair with a number is defined as follows:{{sfn|Jain|2001|p=[https://books.google.com/books?id=-lzAee3uQtIC&pg=PA11 11]}}
<math display="block">
 \begin{align}
 (x_1 , y_1) + (x_2 , y_2) &= (x_1 + x_2, y_1 + y_2), \\
 a(x, y) &= (ax, ay).
\end{align}
</math>

The first example above reduces to this example if an arrow is represented by a pair of [[Cartesian coordinates]] of its endpoint.

===Coordinate space===
The simplest example of a vector space over a field {{math|''F''}} is the field {{math|''F''}} itself with its addition viewed as vector addition and its multiplication viewed as scalar multiplication. More generally, all [[tuple|{{math|''n''}}-tuples]] (sequences of length {{math|''n''}}) 
<math display=block>(a_1, a_2, \dots, a_n)</math>
of elements {{math|''a''<sub>''i''</sub>}} of {{math|''F''}} form a vector space that is usually denoted {{math|''F''<sup>''n''</sup>}} and called a '''coordinate space'''.{{sfn|Lang|1987|loc = ch. I.1}} 
The case {{math|1=''n'' = 1}} is the above-mentioned simplest example, in which the field {{math|''F''}} is also regarded as a vector space over itself. The case {{math|1=''F'' = '''R'''}} and {{math|1=''n'' = 2}} (so '''R'''<sup>2</sup>) reduces to the previous example.

===Complex numbers and other field extensions===
The set of [[complex numbers]] {{math|'''C'''}}, numbers that can be written in the form {{math|1=''x'' + ''iy''}} for [[real numbers]] {{math|''x''}} and {{math|''y''}} where {{math|''i''}} is the [[imaginary unit]], form a vector space over the reals with the usual addition and multiplication: {{math|1=(''x'' + ''iy'') + (''a'' + ''ib'') = (''x'' + ''a'') + ''i''(''y'' + ''b'')}} and {{math|1=''c'' ⋅ (''x'' + ''iy'') = (''c'' ⋅ ''x'') + ''i''(''c'' ⋅ ''y'')}} for real numbers {{math|''x''}}, {{math|''y''}}, {{math|''a''}}, {{math|''b''}} and {{math|''c''}}.  The various axioms of a vector space follow from the fact that the same rules hold for complex number arithmetic. The example of complex numbers is essentially the same as (that is, it is ''isomorphic'' to) the vector space of ordered pairs of real numbers mentioned above: if we think of the complex number {{math|''x'' + ''i'' ''y''}} as representing the ordered pair {{math|(''x'', ''y'')}} in the [[complex plane]] then we see that the rules for addition and scalar multiplication correspond exactly to those in the earlier example.

More generally, [[field extension]]s provide another class of examples of vector spaces, particularly in algebra and [[algebraic number theory]]: a field {{math|''F''}} containing a [[Field extension|smaller field]] {{math|''E''}} is an {{math|''E''}}-vector space, by the given multiplication and addition operations of {{math|''F''}}.{{sfn|Lang|2002|loc = ch. V.1}} For example, the complex numbers are a vector space over {{math|'''R'''}}, and the field extension <math>\mathbf{Q}(i\sqrt{5})</math> is a vector space over {{math|'''Q'''}}.  <!--A particularly interesting type of field extension in [[number theory]] is {{math|'''Q'''(''α'')}}, the extension of the rational numbers {{math|'''Q'''}} by a fixed complex number {{math|''α''}}. {{math|'''Q'''(''α'')}} is the smallest field containing the rationals and a fixed complex number ''α''. Its dimension as a vector space over {{math|'''Q'''}} depends on the choice of {{math|''α''}}.-->

===Function spaces===
{{Main|Function space}}
[[File:Example for addition of functions.svg|class=skin-invert-image|thumb|Addition of functions: the sum of the sine and the exponential function is <math>\sin+\exp:\R\to\R</math> with <math>(\sin+\exp)(x)=\sin(x)+\exp(x)</math>.]]

Functions from any fixed set {{math|Ω}} to a field {{math|''F''}} also form vector spaces, by performing addition and scalar multiplication pointwise. That is, the sum of two functions {{math|''f''}} and {{math|''g''}} is the function <math>(f + g)</math> given by
<math display=block>(f + g)(w) = f(w) + g(w),</math>
and similarly for multiplication. Such function spaces occur in many geometric situations, when {{math|Ω}} is the [[real line]] or an [[interval (mathematics)|interval]], or other [[subset]]s of {{math|'''R'''}}. Many notions in topology and analysis, such as [[continuous function|continuity]], [[integral|integrability]] or [[differentiability]] are well-behaved with respect to linearity: sums and scalar multiples of functions possessing such a property still have that property.{{sfn|Lang|1993|loc = ch. XII.3., p. 335}} Therefore, the set of such functions are vector spaces, whose study belongs to [[functional analysis]].

===Linear equations===
{{Main|Linear equation|Linear differential equation|Systems of linear equations}}
Systems of [[homogeneous linear equation]]s are closely tied to vector spaces.{{sfn|Lang|1987|loc = ch. VI.3.}} For example, the solutions of 
<math display=block>\begin{alignat}{9}
  && a \,&&+\, 3 b \,&\, + &\,   & c & \,= 0 \\
4 && a \,&&+\, 2 b \,&\, + &\, 2 & c & \,= 0 \\
\end{alignat}</math>
are given by triples with arbitrary <math>a,</math> <math>b = a / 2,</math> and <math>c = -5 a / 2.</math> They form a vector space: sums and scalar multiples of such triples still satisfy the same ratios of the three variables; thus they are solutions, too. [[matrix (mathematics)|Matrices]] can be used to condense multiple linear equations as above into one vector equation, namely
<div id=equation3><math display=block>A \mathbf{x} = \mathbf{0},</math></div>
where <math>A = \begin{bmatrix}
1 & 3 & 1 \\
4 & 2 & 2\end{bmatrix}</math> is the matrix containing the coefficients of the given equations, <math>\mathbf{x}</math> is the vector <math>(a, b, c),</math> <math>A \mathbf{x}</math> denotes the [[matrix product]], and <math>\mathbf{0} = (0, 0)</math> is the zero vector. In a similar vein, the solutions of homogeneous ''linear differential equations'' form vector spaces. For example,
<div id=equation1><math display=block>f^{\prime\prime}(x) + 2 f^\prime(x) + f(x) = 0</math></div>
yields <math>f(x) = a e^{-x} + b x e^{-x},</math> where <math>a</math> and <math>b</math> are arbitrary constants, and <math>e^x</math> is the [[natural exponential function]].