Dr. Mark V. Sapir
One of the main ideas of algebra is the following. Consider a set of objects
studied in some area of mathematics or physics or any other science
(say, the set of all numbers, the set of all vectors on a plane, the set of all
functions, the set of all theorems in a calculus book, etc.). Usually there
are certain operations that one can perform on these objects (say, one can add,
multiply, subtract numbers, vectors or functions). A collection of objects
and operations is usually called an algebraic system.
Then we choose some
minimal set of basic operations such that all other useful operations are
composed from the basic ones.
Then we look what kind of important
statements about our system of objects we can prove. We look at the proofs
and find a minimal system of basic properties of our operations,
that we use in these proofs.
After that we say that every algebraic system which satisfies these
basic properties is similar to the one we started with.
And indeed, all important theorems which hold in the initial algebraic
system will hold in any similar algebraic system. The important fact is
that even if two algebraic systems are similar from the algebraic point
of view, they may come from completely different parts of science, and
their nature may be completely different. This allows us to transfer
knowledge from one part of science to another.
For example, consider the set of all vectors on a plane. Choose a coordinate system with two unit orthogonal vectors e and f. Then every vector v on the plane is a linear combination of e and f that is
Using these basic operations we can express many other operations used in geometry. For example, from geometry we know another formula for the dot product:
We can also find the area of a triangle. Of course, first we need to
define a triangle in terms of vectors. An appropriate definition could
be the following: a triangle is a triple of
vectors (a, b, a-b). From school, you remember the formula for the area of a triangle:
Since we know how to compute the angle between two vectors using only the dot product, we can express the area in terms of the dot product.
Thus, we can see that using only the basic operations (addition,
multiplication by scalar, dot product) we can define many important
geometric concepts and operations.
Now we shall consider four other examples of algebraic systems.
One can view a 2-vector (x,y) as just another notation for a vector on
a plane (xe+yf) or as a notation for the complex number x+iy.
We can add two vectors and multiply a vector by a scalar (as matrices). We
can also add two continuous functions (matrices, complex numbers)
and multiply a function (matrix, complex number) by a scalar (=real number).
Since n-vectors are matrices, they satisfy the following properties:
Continuous functions, matrices and complex numbers
satisfy the same properties. In particular,
the zero function is the function which takes every number to zero.
Using these properties we can deduce other properties of vectors, functions,
matrices or complex numbers. For example, in order to solve an equation
Then we can use the associative law:
Then we remember that A=1*A and rewrite the equality in the following way:
Then we use one of the distributivity laws:
Since 1+(-1)=0, 0*A=0=0*X, we can rewrite the equality again:
Now X=1*X and we use one of the distributive laws again:
Since 1+0=1 and 1*X=X, we finally get:
Of course, we denote B+(-A) as B-A, so the subtraction is a derived operation, we derive it from the addition and the multiplication by
scalar. Notice that in the solution of the equation X+A=B we did not use
the fact that A,B,X are functions or vectors (or matrices or numbers for that
matter), we used only the properties of our basic operations.
Any set of objects V where addition and scalar multiplication are defined and satisfy properties 1--7 is called a vector space. Here by addition we mean any operation which associates with each pair of objects A and B from V another object (the sum) C also from V; by a scalar multiplication we mean any operation which associates with every scalar k and every object A from V another object from V called the scalar multiple of A and denoted by kA.
Elements of general vector spaces are usually called vectors. For any system of vectors A1,...,An and for any system of numbers a1,...,an one can define a linear combination of A1,...,An with coefficients a1,...,an as
Notice that a vector space does not necessarily consist of n-vectors or vectors on the plane. The set of continuous functions, the set of k by n matrices, the set of complex numbers are examples of vector spaces.
Not every set of objects with addition and scalar multiplication
is a vector space. For example, we can define the following operations
on the set of 2-vectors:
Scalar multiplication: k(a,b)=(k2a,b).
Then the resulting algebraic system
will not be a vector space because if we take k=3, m=2, a=1, b=1
Thus the third property of vector spaces does not hold.
We can also define dot products in other vector spaces considered in the previous section. The most important one is the dot product of functions. If f and g are two continuous functions on the interval [0,1] then the dot (inner) product f*g is the integral of the product f(x)g(x) from 0 to 1. The dot product of functions f(x) and g(x) will be denoted by <f(x),g(x)>.
The dot product of n-vectors satisfies the following properies:
The dot product of functions from C[0,1] satisfies the same properties.
Any vector space V with a dot product which satisfies properties 1-4 is called
a Euclidean vector space.
Using the dot product one can define most of the geometric concepts, so one can transfer the elementary geometry to arbitrary Euclidean vector spaces.
In particular, one can define the length (norm) of a vector in a vector space as
Theorem. Let V be a Euclidean vector space then the norm has the following properties:
All these properties have clear geometric meanings in the planar
The first property means that length is always non-negative
and the zero vector is the only vector of length 0.
The second property means that if we multiply a vector by a number k,
the vector gets longer by a factor of |k| (if k is negative, the vector
changes its direction).
The third property (the Cauchy-Schwartz inequality) means that
<A,B>/||A|| ||B|| is always between -1 and 1, which is true for vectors on the
plane since this quotient is precisely the cosine of the angle between these
The fourth property means that the length of every side of a triangle
does not exceed the sum of the lengths of the other two sides.
Using the norm, one can define the distance between two vectors:
This distance satisfies the ordinary property of distances:
Theorem. Let V be a Euclidean vector space then the distance function has the following properties:
We can define many other geometric concepts using the dot product. For example, we can call two vectors A and B orthogonal
if <A,B>=0 (their dot product is 0). Orthogonal vectors in arbitrary Euclidean
vector spaces have properties similar to orthogonal vectors on a plane. For
example, if A and B are orthogonal then pA and qB are also orthogonal for
every scalars p and q (
The following theorem is an analogue of the Pythagoras theorem.
Theorem. Let A1,...,An be pairwise orthogonal vectors in a Euclidean vector space. Then
The proof is left as an exercise.
for a discussion of a generalization of Pythagoras theorem to
sets of vectors.
Using dot products in Rn we can rewrite any system of linear equations
in the following form:
v is the vector of unknowns (x1,...,xn)
and * denotes the dot product. This gives another interpretation of systems
of linear equations.
In particular if b1=b2=...=bm=0 then to solve this
system of linear equations means to find a vector v which is orthogonal to
given vectors A1,...,Am. Thus homogeneous systems of linear equations arise
naturally in the geometry of Euclidean vector spaces.