Linear algebra (Osnabrück 2024-2025)/Part I/Lecture 2
A main focus of mathematics is to study how a certain variable (describing a seize or a magnitude) depend on another (or some other) variable (or several variables). For example, how does the area of a square depend on the length of the side, how does the price depend on the commodities bought, how does the size of a population grows with time. Such dependencies are expressed with the concept of a mapping.
Let and denote sets. A mapping from to is given by assigning, to every element of the set , exactly one element of the set . The unique element which is assigned to , is denoted by . For the mapping as a whole, we write
If a mapping is given, then is called the domain (or domain of definition) of the map and is called the codomain (or target range) of the map. For an element , the element
is called the value of at the place (or argument) .
Two mappings and are equal if and only if their domains coincide, their codomains coincide and if for all the equality in holds. So the equality of mappings is reduced to the equalities of elements in a set. Mappings are also called functions. However, we will usually reserve the term function for mappings where the codomain is a number set like the real numbers .
For every set , the mapping
which sends every element to itself, is called the identity (on ). We denote it by . For another set and a fixed element , the mapping
which sends every element to the constant value , is called the constant mapping (with value ). It is usually again denoted by .[1]
There are several ways to describe a mapping, like value table, bar chart, pie chart, arrow diagram, the graph of the mapping. In mathematics, a mapping is most often given by a mapping rule, which allows computing the values of the mapping for every argument. Such rules are e.g. (from to ) , , etc. In the sciences and in sociology also empirical functions are important which describe real movements or developments. But also for such functions, one wants to know whether they can be described (approximated) in mathematical manner.
Examples of such mapping rules are (from to ) , , etc.
- Injective and surjective mappings
Let and denote sets, and let
be a mapping. Then is called injective, if for two different elements , also and
are different.Let and denote sets, and let
be a mapping. Then is called surjective, if for every , there exists at least one element , such that
Let and denote sets and suppose that
is a mapping. Then is called bijective if is injective as well as
surjective.This concepts are fundamental!
The question, whether a mapping has the properties of being injective or surjective, can be understood looking at the equation
(in the two variables and ). The surjectivity means that for every there exists at least one solution
for this equation, the injectivity means that for every there exist at most one solution for this equation, and the bijectivity means that for every there exists exactly one solution for this equation. Hence surjectivity means the existence of solutions, injectivity means the uniqueness of solutions. Both questions are everywhere in mathematics and they also can be interpreted as surjectivity or injectivity of suitable mappings.
In order to show that a certain mapping is injective, we often us the following strategy: One shows for any two given elements and using the condition that holds. This method is often easier than to show that implies .
The mapping
is neither injective nor surjective. It is not injective, because the different numbers and are both sent to . It is not surjective, because only nonnegative elements are in the image (a negative number does not have a real square root). The mapping
is injective, but not surjective. The injectivity can be seen as follows: If , then one number is larger, say
But then also , and in particular . The mapping
is not injective, but surjective, since every nonnegative real number has a square root. The mapping
is injective and surjective.
Let denote a bijective mapping. Then the mapping
which sends every element to the uniquely determined element with ,
is called the inverse mapping of .The inverse mapping is usually denoted by .
We discuss two classes of mappings which are in the framework of linear algebra very important. They are both so-called linear mappings.
It is a goal of linear algebra to determine, in dependence of the entries , whether the mapping defined by the matrix is injective, surjective or bijective, and how, in the bijective case, the inverse mapping looks like.
A healthy breakfast starts with a fruit salad. The following table shows how much vitamin C, calcium and magnesium various fruits have (in milligram with respect to 100 gram of the fruit).
apple | orange | grapes | banana | |
---|---|---|---|---|
vitamin C | ||||
calcium | ||||
magnesium |
This table yields a mapping, which assigns to a -tuple , representing the used fruits, the content of the resulting fruit salad with respect to vitamin C, calcium and magnesium in the form of a -tuplel . This mapping can be described with the matrix
using matrix multiplication as
- Composition of mappings
Let and denote sets, let
and
be mappings. Then the mapping
is called the composition of the mappings
and .So we have
where the left hand side is defined by the right hand side. If both mappings are given by functional expressions, then the composition is realized by plugging in the first term into the variable of the second term (and to simplify the expression if possible).
For a bijective mapping , the inverse mapping is characterized by the conditions
and
The composition of
and
is given by
However,
Hence the composition of two mappings depends on the ordering.
Two mappings are the same if and only if the equality holds for every . So let . Then
- Graph, image and preimage of a mapping
Mapping/Graph/Image/Preimage/Introduction/Section
Mapping/Squaring/Image and preimage/Example
For two given sets and , we denote the set of mappings from to by
- Binary operations
The natural addition assigns to two real numbers another real number, its structure is
Such binary operations play an important role in mathematics.
An operation (or binary operation) on a set is a mapping
A binary operation assigns to a pair
another element
Many mathematical constructions are captured by this concept: addition, difference, multiplication, division of numbers, the composition of mappings, the intersection or the union of sets, etc. Basically any symbol can be used to denote a binary operation, like . Depending on the symbol, we call the binary operation also multiplication or addition, but this does not mean that we are refereing to any natural multiplication. Important structural properties of a binary operation are listed in the following definitions.
Let a set and a binary operation
be given. An element is called neutral element of the operation, if for all the equalities
hold.In the commutative case, it is enough to check only one property of the neutral element.
Let a set with a binary operation
and a neutral element be given. For an element , an element is called inverse element (for ), if the equalities
- Footnotes
- ↑ Hilbert has said that the art of denotation in mathematics is to use the same symbol for different things.
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