Linear Algebra/Sets, Functions, Relations

Linear Algebra
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SetsEdit

Mathematicians work with collections called sets. A set can be given as a listing between curly braces as in  , or, if that's unwieldy, by using set-builder notation as in   (read "the set of all   such that \ldots"). We name sets with capital roman letters as with the primes  , except for a few special sets such as the real numbers  , and the complex numbers  . To denote that something is an element (or member) of a set we use " ", so that   while  .

What distinguishes a set from any other type of collection is the Principle of Extensionality, that two sets with the same elements are equal. Because of this principle, in a set repeats collapse   and order doesn't matter  .

We use " " for the subset relationship:   and " " for subset or equality (if   is a subset of   but   then   is a proper subset of  ). These symbols may be flipped, for instance  .

Because of Extensionality, to prove that two sets are equal  , just show that they have the same members. Usually we show mutual inclusion, that both   and  .

Set operationsEdit

Venn diagrams are handy here. For instance,   can be pictured

 

and " " looks like this.

 

Note that this is a repeat of the diagram for "if \ldots then ..." propositions. That's because " " means "if   then  ".

In general, for every propositional logic operator there is an associated set operator. For instance, the complement of   is  

 

the union is  

 

and the intersection is  

 

}}When two sets share no members their intersection is the empty set  , symbolized  . Any set has the empty set for a subset, by the "vacuously true" property of the definition of implication.

SequencesEdit

We shall also use collections where order does matter and where repeats do not collapse. These are sequences, denoted with angle brackets:  . A sequence of length   is sometimes called an ordered pair and written with parentheses:  . We also sometimes say "ordered triple", "ordered  -tuple", etc. The set of ordered  -tuples of elements of a set   is denoted  . Thus the set of pairs of reals is  .

FunctionsEdit

We first see functions in elementary Algebra, where they are presented as formulas (e.g.,  ), but progressing to more advanced Mathematics reveals more general functions— trigonometric ones, exponential and logarithmic ones, and even constructs like absolute value that involve piecing together parts— and we see that functions aren't formulas, instead the key idea is that a function associates with its input   a single output  .

Consequently, a function or map is defined to be a set of ordered pairs   such that   suffices to determine  , that is: if   then   (this requirement is referred to by saying a function is well-defined).\footnote{More on this is in the section on isomorphisms}

Each input   is one of the function's arguments and each output   is a value. The set of all arguments is  's domain and the set of output values is its range. Usually we don't need know what is and is not in the range and we instead work with a superset of the range, the codomain. The notation for a function   with domain   and codomain   is  .

 

We sometimes instead use the notation  , read "  maps under   to  ", or "  is the "image' of  '.

Some maps, like  , can be thought of as combinations of simple maps, here,   applied to the image of  . The composition of   with  , is the map sending   to  . It is denoted  . This definition only makes sense if the range of   is a subset of the domain of  .

Observe that the identity map   defined by   has the property that for any  , the composition   is equal to  . So an identity map plays the same role with respect to function composition that the number   plays in real number addition, or that the number   plays in multiplication.

In line with that analogy, define a left inverse of a map   to be a function   such that   is the identity map on  . Of course, a right inverse of   is a   such that   is the identity.

A map that is both a left and right inverse of   is called simply an inverse. An inverse, if one exists, is unique because if both   and   are inverses of   then   (the middle equality comes from the associativity of function composition), so we often call it "the" inverse, written  . For instance, the inverse of the function   given by   is the function   given by  .

The superscript " " notation for function inverse can be confusing— it doesn't mean  . It is used because it fits into a larger scheme. Functions that have the same codomain as domain can be iterated, so that where  , we can consider the composition of   with itself:  , and  , etc.

Naturally enough, we write   as   and   as  , etc. Note that the familiar exponent rules for real numbers obviously hold:   and  . The relationship with the prior paragraph is that, where   is invertible, writing   for the inverse and   for the inverse of  , etc., gives that these familiar exponent rules continue to hold, once   is defined to be the identity map.

If the codomain   equals the range of   then we say that the function is onto (or surjective). A function has a right inverse if and only if it is onto (this is not hard to check). If no two arguments share an image, if   implies that  , then the function is one-to-one (or injective). A function has a left inverse if and only if it is one-to-one (this is also not hard to check).

By the prior paragraph, a map has an inverse if and only if it is both onto and one-to-one; such a function is a correspondence. It associates one and only one element of the domain with each element of the range (for example, finite sets must have the same number of elements to be matched up in this way). Because a composition of one-to-one maps is one-to-one, and a composition of onto maps is onto, a composition of correspondences is a correspondence.

We sometimes want to shrink the domain of a function. For instance, we may take the function   given by   and, in order to have an inverse, limit input arguments to nonnegative reals  . Technically,   is a different function than  ; we call it the restriction of   to the smaller domain.

A final point on functions: neither   nor   need be a number. As an example, we can think of   as a function that takes the ordered pair   as its argument.

RelationsEdit

Some familiar operations are obviously functions: addition maps   to  . But what of " " or " "? We here take the approach of rephrasing " " to "  is in the relation  ". That is, define a binary relation on a set   to be a set of ordered pairs of elements of  . For example, the   relation is the set  ; some elements of that set are  ,  , and  .

Another binary relation on the natural numbers is equality; this relation is formally written as the set  .

Still another example is "closer than  ", the set  . Some members of that relation are  ,  , and  . Neither   nor   is a member.

Those examples illustrate the generality of the definition. All kinds of relationships (e.g., "both numbers even" or "first number is the second with the digits reversed") are covered under the definition.

Equivalence RelationsEdit

We shall need to say, formally, that two objects are alike in some way. While these alike things aren't identical, they are related (e.g., two integers that "give the same remainder when divided by  ").

A binary relation   is an equivalence relationwhen it satisfies

  1. reflexivity: any object is related to itself;
  2. symmetry: if   is related to   then   is related to  ;
  3. transitivity: if   is related to   and   is related to   then   is related to  .

(To see that these conditions formalize being the same, read them again, replacing "is related to" with "is like".)

Some examples (on the integers): " " is an equivalence relation, " " does not satisfy symmetry, "same sign" is a equivalence, while "nearer than  " fails transitivity.

PartitionsEdit

In "same sign"   there are two kinds of pairs, the first with both numbers positive and the second with both negative. So integers fall into exactly one of two classes, positive or negative.

A partition of a set   is a collection of subsets   such that every element of   is in one and only one  :  , and if   is not equal to   then  . Picture   being decomposed into distinct parts.

 

Thus, the first paragraph says "same sign" partitions the integers into the positives and the negatives.

Similarly, the equivalence relation "=" partitions the integers into one-element sets.

Another example is the fractions. Of course,   and   are equivalent fractions. That is, for the set  , we define two elements   and   to be equivalent if  . We can check that this is an equivalence relation, that is, that it satisfies the above three conditions. With that,   is divided up into parts.

 

Before we show that equivalence relations always give rise to partitions, we first illustrate the argument. Consider the relationship between two integers of "same parity", the set   (i.e., "give the same remainder when divided by  "). We want to say that the natural numbers split into two pieces, the evens and the odds, and inside a piece each member has the same parity as each other. So for each   we define the set of numbers associated with it:  . Some examples are  , and  , and  . These are the parts, e.g.,   is the odds.


}}Theorem An equivalence relation induces a partition on the underlying set.

Proof

Call the set   and the relation  . In line with the illustration in the paragraph above, for each   define  .

Observe that, as   is a member if  , the union of all these sets is  . So we will be done if we show that distinct parts are disjoint: if   then  . We will verify this through the contrapositive, that is, we wlll assume that   in order to deduce that  .

Let   be an element of the intersection. Then by definition of   and  , the two   and   are members of  , and by symmetry of this relation   and   are also members of  . To show that   we will show each is a subset of the other.

Assume that   so that  . Use transitivity along with   to conclude that   is also an element of  . But   so another use of transitivity gives that  . Thus  . Therefore   implies  , and so  .

The same argument in the other direction gives the other inclusion, and so the two sets are equal, completing the contrapositive argument.

}}We call each part of a partition an equivalence class (or informally, "part").

We somtimes pick a single element of each equivalence class to be the class representative.

 

Usually when we pick representatives we have some natural scheme in mind. In that case we call them the canonical representatives.

An example is the simplest form of a fraction. We've defined   and   to be equivalent fractions. In everyday work we often use the "simplest form" or "reduced form" fraction as the class representatives.

 

Linear Algebra
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