**Linear Maps and Matrices**

The relation of two vector spaces can be expressed by *linear map* or *linear transformation*. They are functions that reflect the vector space structure—i.e., they preserve sums and scalar multiplication:

*ƒ*(**x**+**y**) =*ƒ*(**x**) +*ƒ*(**y**) and*ƒ*(*a*·**x**) =*a*·*ƒ*(**x**) for all**x**and**y**in*V*, all*a*in*F*.

An *isomorphism* is a linear map *ƒ* : *V* → *W* such that there exists an inverse map *g* : *W* → *V*, which is a map such that the two possible compositions *ƒ* ∘ *g* : *W* → *W* and *g* ∘ *ƒ* : *V* → *V* are identity maps. Equivalently, *ƒ* is both one-to-one (injective) and onto (surjective). If there exists an isomorphism between *V* and *W*, the two spaces are said to be *isomorphic*; they are then essentially identical as vector spaces, since all identities holding in *V* are, via *ƒ*, transported to similar ones in *W*, and vice versa via *g*.

For example, the "arrows in the plane" and "ordered pairs of numbers" vector spaces in the introduction are isomorphic: a planar arrow **v** departing at the origin of some (fixed) coordinate system can be expressed as an ordered pair by considering the *x*- and *y*-component of the arrow, as shown in the image at the right. Conversely, given a pair (*x*, *y*), the arrow going by *x* to the right (or to the left, if *x* is negative), and *y* up (down, if *y* is negative) turns back the arrow **v**.

Linear maps *V* → *W* between two fixed vector spaces form a vector space Hom_{F}(*V*, *W*), also denoted L(*V*, *W*). The space of linear maps from *V* to *F* is called the *dual vector space*, denoted *V*∗. Via the injective natural map *V* → *V*∗∗, any vector space can be embedded into its *bidual*; the map is an isomorphism if and only if the space is finite-dimensional.

Once a basis of *V* is chosen, linear maps *ƒ* : *V* → *W* are completely determined by specifying the images of the basis vectors, because any element of *V* is expressed uniquely as a linear combination of them. If dim *V* = dim *W*, a 1-to-1 correspondence between fixed bases of *V* and *W* gives rise to a linear map that maps any basis element of *V* to the corresponding basis element of *W*. It is an isomorphism, by its very definition. Therefore, two vector spaces are isomorphic if their dimensions agree and vice versa. Another way to express this is that any vector space is *completely classified* (up to isomorphism) by its dimension, a single number. In particular, any *n*-dimensional *F*-vector space *V* is isomorphic to *F**n*. There is, however, no "canonical" or preferred isomorphism; actually an isomorphism φ: *F**n* → *V* is equivalent to the choice of a basis of *V*, by mapping the standard basis of *F**n* to *V*, via φ. The freedom of choosing a convenient basis is particularly useful in the infinite-dimensional context, see below.

Read more about this topic: Vector Space

### Famous quotes containing the word maps:

“And at least you know

That *maps* are of time, not place, so far as the army

Happens to be concerned—the reason being,

Is one which need not delay us.”

—Henry Reed (1914–1986)