# Standard base

In the mathematical sub-area of linear algebra, a standard basis , natural basis , unit basis or canonical basis is a special basis that is already distinguished from all possible bases in certain vector spaces due to its construction.

## General basis

In general, a basis of a vector space is a family of vectors with the property that each vector of the space can be clearly represented as a finite linear combination of these. The coefficients of this linear combination are called the coordinates of the vector with respect to this base. One element of the basis is called a basis vector.

Every vector space has a basis, and in general numerous bases, none of which is distinguished.

### Examples

• The parallel displacements of the plane of observation form a vector space ( see Euclidean space ) of dimension two. However, no base is excellent. A possible basis would consist of a “shift by one unit to the right” and the “shift by one unit up”. Here, "unit", "right" and "above" are conventions or depending on the perspective.
• Those real-valued functions which are twice differentiable and which satisfy the equation for all form a real vector space of dimension two. A possible basis is formed by the sine and cosine functions. Choosing this base may seem like an obvious choice, but it doesn't stand out very well over other choices.${\ displaystyle f \ colon \ mathbb {R} \ to \ mathbb {R}}$${\ displaystyle x \ in \ mathbb {R}}$${\ displaystyle f (x) + f '' (x) = 0}$${\ displaystyle V}$

## Standard basis in the standard rooms

Standard basis vectors in the Euclidean plane

The vector spaces that are usually introduced first are the standard spaces with . Elements of are all - tuples of real numbers. From all bases of the one can distinguish the one with respect to which the coordinates of a vector coincide exactly with its tuple components. So this basis consists of where ${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle n \ in \ mathbb {N}}$${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle n}$ ${\ displaystyle \ mathbb {R} ^ {n}}$${\ displaystyle e_ {1}, \ ldots, e_ {n}}$

${\ displaystyle {\ begin {matrix} e_ {1} & = & (1,0,0, \ ldots, 0), \\ e_ {2} & = & (0,1,0, \ ldots, 0) , \\ & \ vdots & \\ e_ {n} & = & (0,0,0, \ ldots, 1) \ end {matrix}}}$

and is referred to as the standard base of the . ${\ displaystyle \ mathbb {R} ^ {n}}$

The same applies to the vector space over an arbitrary body , that means here too there are the standard basis vectors . ${\ displaystyle K ^ {n}}$ ${\ displaystyle K}$${\ displaystyle e_ {1} = (1.0, \ ldots, 0), \ ldots, e_ {n} = (0, \ ldots, 0.1)}$

### example

The standard basis of the consists of and . The two vector spaces listed above as examples are isomorphic to , but do not have a standard basis. As a result, even among the isomorphisms between these spaces and none is distinguished. ${\ displaystyle \ mathbb {R} ^ {2}}$${\ displaystyle e_ {1} = (1,0)}$${\ displaystyle e_ {2} = (0.1)}$${\ displaystyle \ mathbb {R} ^ {2}}$${\ displaystyle \ mathbb {R} ^ {2}}$

### designation

The designation for the standard basis vectors is widely used. However, the three standard basis vectors of three-dimensional vector space are sometimes referred to in the applied natural sciences as : ${\ displaystyle e_ {1}, e_ {2}, \ ldots}$${\ displaystyle \ mathbb {R} ^ {3}}$${\ displaystyle \ mathbf {i}, \, \ mathbf {j}, \, \ mathbf {k}}$

${\ displaystyle \ mathbf {i} = e_ {1} = {\ begin {pmatrix} 1 \\ 0 \\ 0 \ end {pmatrix}}, \ quad \ mathbf {j} = e_ {2} = {\ begin {pmatrix} 0 \\ 1 \\ 0 \ end {pmatrix}}, \ quad \ mathbf {k} = e_ {3} = {\ begin {pmatrix} 0 \\ 0 \\ 1 \ end {pmatrix}}}$

### Other properties

It has other properties beyond the vector space property. With regard to this, too, the standard basis vectors often meet special conditions. So the standard basis is an orthonormal basis with respect to the standard scalar product . ${\ displaystyle \ mathbb {R} ^ {n}}$

## Standard base in the die space

The set of matrices over a body also forms a vector space with the matrix addition and the scalar multiplication . The standard basis in this matrix space is formed by the standard matrices in which exactly one entry is equal to one and all other entries are equal to zero. For example, the four form matrices ${\ displaystyle K ^ {m \ times n}}$ ${\ displaystyle E_ {ij}}$

${\ displaystyle E_ {11} = {\ begin {pmatrix} 1 & 0 \\ 0 & 0 \ end {pmatrix}}, E_ {12} = {\ begin {pmatrix} 0 & 1 \\ 0 & 0 \ end {pmatrix}}, E_ {21 } = {\ begin {pmatrix} 0 & 0 \\ 1 & 0 \ end {pmatrix}}, E_ {22} = {\ begin {pmatrix} 0 & 0 \\ 0 & 1 \ end {pmatrix}}}$

the standard base of the space of the matrices. ${\ displaystyle (2 \ times 2)}$

## Standard basis in infinitely dimensional spaces

If a body and an arbitrary (especially possibly infinite) set, the finite formal linear combinations of elements form a vector space. Then this vector space is itself the basis and is called its standard basis. ${\ displaystyle K}$${\ displaystyle M}$${\ displaystyle M}$${\ displaystyle M}$

Instead of formal linear combinations, one also alternatively considers the vector space of those mappings with the property that applies to almost all . To be through ${\ displaystyle f \ colon M \ to K}$${\ displaystyle f (x) = 0}$ ${\ displaystyle x \ in M}$${\ displaystyle m \ in M}$${\ displaystyle e_ {m} \ colon M \ to K}$

${\ displaystyle e_ {m} (x) = {\ begin {cases} 1, & {\ text {falls}} x = m \\ 0, & {\ text {falls}} x \ neq m \ end {cases }}}$

given figure . Then the family forms a basis of the vector space, which in this case is also called the standard basis. ${\ displaystyle M \ to K}$${\ displaystyle \ {e_ {m} \} _ {m \ in M}}$

The vector space of all mappings , on the other hand, has no standard basis if it is infinite. ${\ displaystyle f \ colon M \ to K}$${\ displaystyle M}$

Also polynomial rings over fields are vector spaces where a base has been awarded directly on the basis of the construction. Thus the elements of the polynomial ring are by definition the finite linear combinations of the monomials , etc., which accordingly form a basis - the standard basis - of . ${\ displaystyle \ mathbb {R} [X]}$${\ displaystyle 1,}$ ${\ displaystyle X,}$ ${\ displaystyle X ^ {2},}$ ${\ displaystyle X ^ {3}}$${\ displaystyle \ mathbb {R} [X]}$

## Connection with universal properties

The term canonical is generally used in constructions about a universal property . So there is also a connection between standard bases and the following construction:

Be a body and any quantity. We are looking for a vector space together with a mapping in its underlying set, so that there is exactly one linear mapping for every vector space and every mapping with . Such a pair is then called a canonical mapping or universal solution of with respect to the forgetting function that assigns the underlying set to each vector space. ${\ displaystyle K}$${\ displaystyle M}$${\ displaystyle K}$${\ displaystyle U}$${\ displaystyle f \ colon M \ to U}$${\ displaystyle K}$${\ displaystyle X}$${\ displaystyle g \ colon M \ to X}$ ${\ displaystyle h \ colon U \ to X}$${\ displaystyle g = h \ circ f}$${\ displaystyle (U, f)}$${\ displaystyle f}$${\ displaystyle M}$${\ displaystyle K}$

The vector spaces with standard basis given above have exactly this universal property. The image from under the canonical mapping are precisely the vectors of the canonical base or the canonical mapping as a family is the canonical base. ${\ displaystyle M}$

The fact that there is always such a universal solution exists, already follows that a picture that lots of such a universal solution , and each such maps, a functor , the linksadjungiert for forgetful is. Such a functor is called a free functor . ${\ displaystyle M}$${\ displaystyle U}$${\ displaystyle g}$${\ displaystyle h}$

## literature

• Kowalsky and Michler: Lineare Algebra , Gruyter, ISBN 978-3110179637
• Albrecht Beutelspacher: “That is trivial oBdA!” 9th updated edition, Vieweg + Teubner, Braunschweig and Wiesbaden 2009, ISBN 978-3-8348-0771-7 , sv “Canonical”