Linear Algebra Notes

This page contain's notes on G.E. Shilov's Linear Algebra.

Determinants

Fields

A field (Shilov calls them a number field) is a set KK where addition and multiplication are defined and satisfy the usual arithmetical properties: associativity, commutativity, existance of a "0" and a "1", and distributivity.

The most useful examples of fields for practical applications of linear algebra are the field of real number R\bf R and the field of complex numbers C\bf C. Examples of other kinds of fields include the field of rational numbers and the field of integers modulo a prime numbers.

Systems of Linear Equations

A system of linear equations is an expression of the form

a11x1+a12x2++a1nxn=b1a21x1+a22x2++a2nxn=b2ak1x1+ak2x2++aknxn=bn\begin{aligned} a_{11}x_1 &+ a_{12}x_2 &+ \ldots &+ a_{1 n} x_n = b_1 \\ a_{21}x_1 &+ a_{22} x_2 &+ \ldots &+ a_{2 n} x_n = b_2 \\ \vdots \\ a_{k1}x_1 &+ a_{k2} x_2 &+ \ldots &+ a_{k n} x_n = b_n \end{aligned}

The aa's (coefficients) and bb's (//constant term/) are given quantities and the xx's are "unknown" quantities. Solving this system of equations means finding the set of possible values of xx's (if any) that satisfy each equation. The values of all quantities, given and unknown, are assumed to belong to some given field KK.

In attempting to solve such equations, it may turn out that no such solution exists. Such a system is said to be incompatable. If a solution exists, then the system is classified as determinate if such a solution is the only solutions.

Calculating Determinants

A square matrix is a rectangular arrangement of n2n^2 numbers aijKa_{ij}\in K (i,j=1,,ni,j=1,\ldots,n):

a11a12a1nan1an2ann\begin{matrix} a_{11} & a_{12} & \ldots a_{1 n} \\ \vdots \\ a_{n1} & a_{n2} & \ldots a_{n n} \end{matrix}

(More formally, a square matrix can be thought of as a KK valued function defined on the set of possible indices (i,j)(i, j). )

For defining determinants, Shilov introduces a special function NN that takes in a tuple in Zn\bf Z^n and outputs a natural number. It is defined in a combinatorial way: N(α1,α2,αn)N (\alpha_1, \alpha_2, \dots \alpha_n) is the number of time αi>αj\alpha_i>\alpha_j for i<ji<j. That is, it counts the number of "inversions".

With this notation, the determinant DD of the previously written square matrix is defined as

D=(1)N(α1,,αn)i=1naαi.D = \sum (-1)^{N(\alpha_1, \ldots, \alpha_n)} \prod_{i=1}^{n} a_{\alpha_i}.

Here, the above summation is over all sequences α1,,αn\alpha_1,\ldots,\alpha_n in the set {1,,n}\{1,\ldots,n\} such that αiαj\alpha_i \neq \alpha_j whenever iji \neq j.

The determinant of the matrix AA can be denoted as either detA\det A or as A|A|.

Properties of Determinants

We list some properties of determinants. Denote AA as an n×nn \times n square matrix with elements aija_{ij}. Then:

  1. A square matrix with two identical columns has a zero determinant.
  2. A square matrix with a column consisting of zeroes has a zero determinant.
  3. Let b1,,bnb_1,\ldots,b_n and c1,,cmc_1,\ldots, c_m be sequences of numbers and let λ\lambda be an arbitrary number such that aij=bi+λcia_{ij}=b_{i}+\lambda c_{i} for some given jj. Let BB be the square matrix obtained from AA by replacing the jth column with the column of numbers bib_i. Similarly, let the square matrix CC be obtained from AA by replacing the jth column with the column of numbers cic_{i}. Then detA=detB+λdetC\det A = \det B + \lambda \det C.
  4. The detminant of a matrix is not changed by adding a multiple of values of one column to another column. That is to say, replacing aija_{i j} with aij+λaika_{i j} + \lambda a_{i k} for all ii and some values jj and kjk\neq j will not alter the determinant.
  5. Swapping the rows and columns of a square matrix (the result being called a transposition of the matrix) does not alter the value of the determinant.

As a consequence of the last property, the previously mentioned properties hold true if one were to replace "columns" with "rows".

Minors and Cofactors

TODO

Cramer's Rule

TODO

Laplace's Theorem

Linear Dependence

Linear Spaces


Created 2020-04-08. Last updated 2020-06-22. View source.