A treatise on plane and spherical trigonometry - download pdf or read online

By William Chauvenet

ISBN-10: 1406968250

ISBN-13: 9781406968255

This quantity is made out of electronic photos created in the course of the college of Michigan college Library's maintenance reformatting application.

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Extra resources for A treatise on plane and spherical trigonometry

Sample text

0 () 0 d, ... 0 0 0 ... D = | Special 0 = (dikdk) dm case matrix 0 0 .. ά E = | 0 0 1 . . 0 | = (djfc) Kronecker symbol [ 0 for i \ 1 for φ k 2. ' ^ A; aik Example : fil 2 7\ 0 3 4 J upper triangular matrix ( 0 0 2/ 3 0 0\ 7 4 0 1 lower triangular matrix 9 16/ If certain rows and columns of a matrix A are eliminated to leave a square matrix with k rows a n d k columns of which the determinant can be formed, a k-rowed subdeterminant of A is obtained. 10. Matrices 69 4-rowed subdeterminants 1 2 2 3 12 3 4 2 3 6 5 or 4 0 1 3 1 1 2 4 2 3 5 2 4 0 3 1 etc.

The rank of a matrix is not changed by taking linear combinations of its rows and columns. Example: B y a linear combination of the first and the third columns we obtain All subdeterminants of the third order disappear because three of them only have elements 0 in the first column and the fourth of 70 1. Arithmetic them shows two equal columns as a result of a linear combination of the first and second columns. g. not equal to zero, the matrix has rank k = 2. 2. Theorems on matrices Given: ra,n-matrices A = (a^) and Β = (b t f ).

21 c2 «23 · · «2» «31 c3 «33 ·· «3n «ni Cn «n3 «n3 .. 9. 65 Determinants Δχ«2 χ*3 = — , Δ α 3η Δχο where etc. e. £ = { 3 ; - 2 ; 1 } 5 Bartsch, Mathematical Formulas Δζ •• -2 -1 7 -3 17 -2 12 1. e. if all numerator determinants Δχι disappear. If Δ Φ 0 , there will be only the trivial solution Xl = X% = #3 = · · · = x n = 0 If Δ = 0 , there will be systems of solutions where n o t all variables will disappear. (b) Inhomogeneous system of equations A system of equations is called inhomogeneous if a t least one Cj Φ 0.

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A treatise on plane and spherical trigonometry by William Chauvenet

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