IDA: Prerequisites

Prerequisites

Algebraic identities

Equivalence relations

Functions and Maps

Graphs

Induction

Induction

Matrices

Number systems

Quadratic equations

Sets

Vector spaces

Newton's binomium refers to the identity

(a + b)ⁿ = aⁿ + binom(n,1)a^{n - 1}b + binom(n,2)a^{n - 2}b² + binom(n,3)a^{n - 3}b³ + · · · + binom(n,n -1)a¹b^{n - 1} + bⁿ.

Here, binom(n,k) stands for the binomial coefficient

(ⁿ_k) = n!/(k! (n - k)!)

(`n choose k'), and n is a positive integer. The identity holds for complex numbers a and b, but is also valid for more general arithmetic systems as Algebra Interactive illustrates.

Special cases occur for small values of the exponent:

(a + b)² = a² + 2ab + b² and (a + b)³ = a³ + 3a²b + 3ab² + b³.

Here are some more useful identities valid for complex numbers.

a² - b² = (a + b)(a - b).
a³ - b³ = (a - b)(a² + ab + b²).
(For positive integers n.)
aⁿ - bⁿ = (a - b)(a^{n - 1} + a^{n - 2}b + a^{n - 3}b³ + · · · + b^{n - 1})
with special case (take b = 1)
aⁿ - 1 = (a - 1)(a^{n - 1} + a^{n - 2} + · · · + a + 1).

Given an equivalence relation R on a set A, the set A can be partitioned into subsets - called equivalence classes - as follows. The elements a and b belong to the same equivalence class if and only if (a, b)

R. The equivalence class containing an element a consists precisely of all elements of A equivalent with a.

An equivalence relation is a relation R on a set A which is reflexive, symmetric and transitive. If (a,b)

R, then we say that a and b are equivalent (with respect to R).

A relation R on a set X is called reflexive if (x,x)

R for all x

A relation R on a set X is called symmetric if (x,y)

R implies (y,x)

A relation R on a set X is called transitive if (x,y)

R and (y,z)

R imply (x,z)

A relation on a set V is a subset of V × V. To indicate that the pair (a, b) belongs to this subset, it is common practice to write that a is related to b, or to replace the words `is related to' by some symbol like ~ or by an abbrevation of the relation being discussed.

A map f : A -> B is bijective if it is injective and surjective. This is equivalent to saying that f has an inverse, i.e., there exists a map g say, such that

f(g(b)) = b for all b

B and g(f(a)) = a for all a

If f is a bijection, then its inverse is unique and often denoted by f^-1.

If f : A -> B is a map with codomain B and g : B -> C is a map with domain B, then the composition of f and g is the map

gf : A -> C given by (gf((a) = g(f(a)).

The composition gf is also denoted by g o f. If n is a positive integer and f : A -> A is a map whose domain and codomain coincide, then fⁿ : A -> A is the map obtained by composing f exactly n times with itself. By definition, f⁰ denotes the identity map. For nonnegative integers m and n the equalities

f^{m + n} = f^mfⁿ and (f^m)ⁿ = f^mn

hold.

If f : A -> A is a bijection with inverse f^-1, and if n is a positive integer, then f^-n is defined as (f^-1)ⁿ. For a bijection f : A -> A the rules

f^{m + n} = f^mfⁿ and (f^m)ⁿ = f^mn

hold for all integers m and n.

A function f : A -> R (where in calculus A is usually some open subset of R) is differentiable at a point a

A if the limit

lim_{h -> 0} {f(a + h) - f(a)}/h

exists. In this case the limit is called the derivative of f at a and is usually denoted by f'(a).

A function f : A -> R is differentiable if it is differentiable at every point of A. In that case the derivative f'.

A map f : A -> B is often referred to as a function if B is a set of numbers, for instance the reals. The distinction between maps and functions is not very strict. The term function is more common in calculus.

Given a map f : A -> B and a subset C of B, the image of C under f is the following subset of B:

f(C) = {f(a) | a

C}.

The image of an element a A is the element f(a).

A map f : A -> B is called injective if the following holds for all x and y in A:

If x

y then f(x)

f(y).

In words: Distinct elements of A are mapped to distinct elements of B.

An equivalent formulation: For all x and y in A, the equality f(x) = f(y) implies x = y.

If the f : A -> B is bijective then there exists a unique map g : B -> A such that

f(g(b)) = b for all b

B and g(f(a)) = a for all a

The map g is the inverse of f and is usually denoted by f^-1.

A map f : A -> B (`the map f from A to B') assigns to each element a of A a unique element f(a) of B. The element f(a) is the image of a under f.

We often simply write f instead of f : A -> B. The set A is the domain of the map, the set B the codomain.

Given a map f : A -> B and a subset C of B, the preimage of C under the map f is the following subset of A:

f^-1(C) = {a

A | f(a)

B}.

Note that the notation f^-1 does not refer to the inverse of f (f need not even have an inverse). The reason for the common notation for two different notions is that they are closely related for bijections. If f is a bijection, with inverse denoted by g rather than by f^-1 for the sake of clarity, then for b B we have

{g(b)} = f^-1({b}).

In fact, for every subset C of B the image g(C) equals the preimage f^-1({C}).

The preimage of an element b B is the preimage of {b}. By abuse of notation this preimage is also often written as f^-1(b) rather than f^-1({b}).

A map f : A -> B is called surjective (or onto) if for every b

B there exists an element a

A such that f(a) = b.

Equivalently, in terms of the image of the map, f is surjective if and only if f(A) = B.

A cycle in a graph is a path whose begin and end point coincide.

A graph is called connected if any two vertices are connected by a path.

A graph (without loops) is a set X together with a collection E of subsets of size 2. Members of X are called vertices, members of E are called edges.

Graphs are drawn by representing the vertices by points (or small circles) and edges by lines (or curves) connecting the two vertices of which it consists.

A path in a graph is a sequence of edges such that two consecutive edges have exactly one vertex in common; moreover, the first edge has a vertex (not appearing in the second edge) called begin point, and the last edge has a vertex (not appearing in the one but last edge; if there is just one edge: the non-begin point) called end point. One can think of a path as a walk from begin to end point along edges.

A tree is a connected graph without cycles. A rooted tree is a tree with a distinguished vertex, the root.

If a and b are vertices of a tree, then there is a unique path from a to b: Since the tree is connected, there is a path from a to b; since there are no cycles, the path is unique.

Induction (or mathematical induction) can often be used to prove a property P(n) that is to hold for every natural number n. A proof by induction consists of two steps.

A proof of P(0).
A proof that, for all n, if P(n) holds the property P(n + 1) also holds.

Together these two steps provides a proof that the property P(n) holds for all natural numbers.

There are several variations of `proof by induction'. For instance, if a property Q(n) is to hold for n = 1, 2, ..., then the first step of the proof by induction consists of course of proving Q(1). Another variation is obtained by replacing the second step above by a proof that for all n the property P(n + 1) is implied by P(0), P(1), ..., P(n).

The characteristic polynomial of a square matrix A is the following polynomial in the indeterminate X:

det(XI - A).

Here, I is the identity matrix and det stands for determinant. If A is an n by n matrix then the characteristic polynomial has degree n.

The determinant is a number computed from a square matrix. If A is an n by n matrix with entries A_i,j, then the determinant det(A) of A is defined by the expression

det(A) =

_g sgn(g) A_1,g(1) A_2,g(2) ··· A_n,g(n),

where g runs through all the permutations of the set {1, 2, ..., n}, and where sgn denotes the sign of a permutation. The notion of a permutation is explained in Chapter 5, as well as the notion sign.

The computation of determinants is facilitated by a range of practical methods that are derived from the definition. In fact, the reader may well have computed determinants without having been aware of the formal definition, since the definition involves permutations, a notion that is not always discussed in elementary linear algebra.

For a 2 by 2 matrix [[a, b], [c, d]], the determinant is ad - bc.

An important property of the determinant is its multiplicativity. If A and B are n by n matrices, then

det(AB) = det(A) · det(B).

A square matrix A is called invertible if there exists a matrix B such that AB = BA = I, where I is the identity matrix of the same size. If the square matrix A is invertible, then the matrix B such that AB = BA = I is unique and has the same size as A. It is called the inverse of A and is usually denoted by A^-1.

A useful criterion for invertibility is that A is invertible if and only if det(A) 0, where det stands for the determinant.

The diagonal of a square matrix A is the set of entries A_{i, i}. A diagonal matrix is a square matrix all of whose entries off the diagonal are 0.

The identity matrix of size n by n is the n by n matrix whose off-diagonal entries are all 0 and whose diagonal entries are all 1. The matrix is often denoted by I_n or simply by I.

A matrix is a rectangular array of numbers. Each entry (i.e., element occurring in the matrix) of the matrix belongs to a (unique) row and column. If A is a matrix, then the entry in the i-th row and j-th column is usually denoted by A_i,j, or by another symbol with subindices i and j. If A has m rows and n columns, then A is said to be an m by n matrix, or to have size m by n. For typographical reasons we sometimes write a matrix as a `list of lists'. For example,

[[1, 2, 3],[4, 5, 6]]

is a 2 by 3 matrix with rows [1, 2, 3] and [4, 5, 6].

The set of m by n matrices whose entries are in the set Q, R, etc., is often denoted by M_{m, n}(Q), etc. If m = n, the notation M_m(Q), etc., is used.

If A is an m by n matrix and B is a n by p matrix, then the product AB is the m by p matrix C whose r,s-th entry is

A_r,1B_1,s + A_r,2B_2,s + ··· + A_r,nB_n,s,

where A has entries A_i,j (for i ranging from 1 to m and j ranging from 1 to n) and B has entries B_i,j (for i ranging from 1 to n and j ranging from 1 to p). So, the entry C_r,s is computed from the r-th row of A and the s-th column of B.

Note that the product AB is only defined if the number of columns of A equals the number of rows of B. In particular, if AB exists then the product of B and A (in that order) need not exist. Moreover, if both AB and BA exist, they need not be equal.

A minimal polynomial of a square matrix A is a nonzero polynomial f of minimal degree such that f(A) = 0 (the zero matrix). Here, f(A), i.e., substitution of A in the polynomial f, is to be interpreted as follows. If

f = a₀ + a₁X + ··· + a_nXⁿ,

then

f(A) = a₀I + a₁A + ··· + a_nAⁿ

(again a square matrix of the same size as A), where I is the identity matrix.

The Cayley-Hamilton theorem from linear algebra states that the characteristic polynomial f_A of the square matrix A has the property f_A(A) = 0. Using division with remainder as explained in Chapter 3, it follows that minimal polynomials exist and that they divide the characteristic polynomial.

Usually, one speaks of the minimal polynomial if the leading coefficient is 1.

A square matrix is a matrix of size n by n, i.e., a matrix where the number of rows is equal to the number of columns.

If A is an m by n matrix, then the transpose A^T of A is the n by m matrix whose r,s-th entry is the s,r-th entry of A.

N stands for the set of natural numbers, i.e., the set {0, 1, 2, 3, 4, ...}.

Z stands for the set of integers, i.e., the set {..., -3, -2, -1, 0, 1, 2, 3, 4, ...}.

Q stands for the set of rational numbers, i.e., the set of numbers of the form a/b, with a and b integers and with nonzero b.

R stands for the set of real numbers.

C stands for the set of complex numbers. Complex numbers are of the form a + bi, where a and b are real numbers and where i² = -1.

Given the quadratic expression x² + bx + c, with b and c complex numbers say, the term completing the square refers to rewriting the expression as

(x + b/2)² -b²/4 + c. The terms x² and bx have been absorbed in a square.

Completing the square is a technique used for solving quadratic equations with complex coefficients, say. If d denotes the (a) square root of b²/4 - c, then it follows that the equation

x² + 2bx + c = 0 has the solutions

-b/2 + d and -b/2 - d.

A quadratic equation with complex coefficients b and c can always be solved within the complex numbers. Solutions of a given equation need not exist in smaller number systems, like the reals or rationals.

The Cartesian product of two sets A and B is the set A × B consisting precisely of all the ordered pairs (a, b) with a

A and b

B. The Cartesian product A₁ × A₂ × ··· × A_n of the sets A₁, A₂, ..., A_n is defined analogously. It consists of the ordered n-tuples (a₁, a₂, ..., a_n) of elements a₁

A₁, ..., a_n

A_n.

The complement A\B of the sets A and B is the set

A\B = { x | x belongs to A but not to B}.

The intersection of two sets V and W is defined as

W = { x | x

V and x

W}.

Similarly, the intersection of more than two sets is defined.

A set is just a collection of things, usually called elements. If a is an element of the set A, we denote this by a

A. The empty set is denoted by ø.

A set A is a subset of the set B if every element of A is also an element of B. If A is a subset of B this is denoted by

If A is a subset of B, we also sometimes say that B is superset of A. Notation:

B A.

The union of two sets V and W is defined as
V W = { x | x V or x W}.
Similarly, the union of more than two sets is defined.

If there exists a finite set of vectors that span a vector space V, then V is said to be finite-dimensional. The vector spaces we consider are finite-dimensional. A subset v₁, ..., v_n is a basis of V if every vector in V can be expressed in a unique way as a linear combination of these vectors.
An equivalent way of saying that v₁, ..., v_n is a basis is that

v₁, ..., v_n span V,
the only solution to the equation
a₁ · v₁ + a₂ · v₂ + ··· + a_n · v_n = 0
in the unknowns a₁, ..., a_n is a₁ = ··· = a_n = 0. (The vectors v₁, ..., v_n are independent.)

A linear combination of the vectors v₁, ..., v_m is a vector that can be written in the form
a₁ · v₁ + a₂ · v₂ + ··· + a_m · v_m,
for some scalars a₁, ..., a_m.

The dimension of a finite-dimensional vector space is the number of elements in a basis of the vector space. This number is independent of the basis chosen.

If V is a vector space and U and W are subspaces of V, then V is the direct sum of U and W if every vector v in V can be written in a unique way as v = u + w for some u in U and some w in W. If V is the direct sum of U and V then this is often denoted by V = U W.

Let V and W be vector spaces over the field K. A linear map (or linear transformation) f : V -> W is a map satisfying
f(a₁v₁ + a₂v₂) = a₁f(v₁) + a₂f(v₂)
for all vectors v₁, v₂ in V and for all scalars a₁, a₂ in K.
The kernel of the linear map f is the linear subspace (of V)
{v V | f(v) = 0}.
The image of f is the linear subspace f(V) of W.
Suppose f : V -> W is a linear map between finite-dimensional vector spaces, and v₁, ..., v_n is a basis of V and w₁, ..., w_m is a basis of W, then the matrix of the linear map f with respect to these bases is the m by n matrix whose k-th column consists of the coordinates of f(v_k) with respect to the basis of W.

Given a subset of a vector space V, its span is the set of all linear combinations of vectors from the subset. The span is a subspace of V, i.e., itself a vector space with the operations from V.

A K-vector space consists of a set V with two operations: Addition (+) and scalar multiplication (* or · or no symbol at all) with scalars from a field K. (The notion field is discussed extensively in chapter 7, but for now think of the rational numbers Q, the real numbers R or the complex numbers C.) The elements of the set V are called vectors. The two operations are required to satisfy the following axioms (for all vectors in V and all scalars; vectors are denoted in boldface).

v + w = w + v (commutativity);
(v + w) + u = v + (w + u) (associativity);
There exists a (unique) zero vector 0 such that v + 0 = v;
For every vector v there exists a (unique) vector -v (the inverse of v) such that v + -v = 0;
a · (v + w) = a · v + a · w (distributivity);
(a + b) · v = a · v + b · v (distributivity);
a(b · v) = (ab) · v;
1 · v = v.

These axioms formalize that computations in a vector space behave as expected. Examples of vector spaces:

The set Rⁿ of real n-tuples with
(a₁, ..., a_n) + (b₁, ..., b_n) = (a₁ + b₁, ..., a_n + b_n) and r ·(a₁, ..., a_n) = (ra₁, ..., ra_n)
as addition and scalar multiplication, respectively, is an R-vector space.
Similarly, Cⁿ is a C-vector space and Qⁿ is a Q-vector space.
If one admits only rational scalars in the example Rⁿ, then Rⁿ becomes a Q-vector space.
The set of maps from a set X to R with pointwise addition and multiplication with reals.