Conservapedia - User contributions [en]

Linear algebra

2011-11-04T14:12:17Z

PatrioticMath:

'''Linear algebra''' is the branch of [[mathematics]] that deals with [[vector]]s, vector spaces, linear maps, and systems of linear [[equation]]s. Topics studied in Linear algebra include Linear equations, Matrices, [[Matrix]] decompositions, Computations, Vectors, Vector spaces, Multilinear algebra, Affine space, Projective space.<ref>[http://www.m-w.com/cgi-bin/dictionary?book=Dictionary&va=linear+algebra Merriam Webster definition of Linear Algebra]</ref>
Linear algebra has numerous applications in [[engineering]], [[chemistry]] and [[physics]]. Matrices can be used for data fitting tasks such as "curving" student grades on exams, and for finding correlations between two sets of data. Linear algebra is particularly useful for organizing and simplifying data.

The three most vexing computations in linear algebra are these:
#linear equations
#least squares ([[data fitting]])
#finding the eigenvalues of an ''n x n'' matrix for n > 3.

== Basic Concepts ==

*[[Linear equations]]
:*coefficient matrices and Gauss-Jordan elimination
:*[[rank]] (row rank = column rank)
:*geometric representations, especially [[vector]]s and systems with more variables than equations

*[[Linear transformations]]
:*transformations, inverses and matrix products

*Subspaces
:*[[image (mathematics)|image]] and [[kernel (geometry)|kernel]]
:*[[basis]] and [[span]]
:*[[dimension]]

*[[Determinant]]s
:*[[Sarrus's rule]]
:*geometrical interpretation
::*classical adjoint, expansion factor, application to [[parallelepiped]]s)
:*determinant of similar matrix, [[inverse matrix]], product of matrices
:*Cramer's rule (with and without product rule)
:*[[minor]] of a matrix
:*Laplace expansion (cofactors)

*[[Basis]]
:*[[eigenbasis]]
:*[[orthogonal]] eigenbasis
:*[[orthonormal]] eigenbasis

*[[Eigenvalue]]s and [[eigenvector]]s
:*diagonalization
:*[[trace]]
:*[[characteristic polynomial]]
:*complex eigenvalues

*[[Orthogonality]]
:*orthogonal (perpendicular) vectors
:*orthonormal vectors
:*orthogonal projections
:*[[orthogonal matrix]]
:*projections
:*Gram-Schmidt Process and [[QR factorization]]
:*orthogonal matrices, orthogonal transformations
:*data fitting, especially least squares

*[[Symmetric matrix]]
:*[[spectral theorem]]

*[[Quadratic forms]]

*Linear dynamical systems
:*Euler's Formula

=== Notation ===

*matrices are commonly represented by ''A'' and ''B''
*diagonal matrices are represented by ''D''
*an upper triangular matrix is represented by ''R'', as in QR factorization
*when ''A'' is similar to ''B'', then an [[invertible matrix]] ''S'' is used to represent that ''AS'' = ''SB''

== More advanced topics include ==

*[[Vector space]]s (linear spaces)
:*the conditions of a vector space
:*isomorphisms
:*Nth dimensional spaces and subspaces

*Inner spaces
:*inner product spaces

*Determinants
:*cofactor
:*adjugate (useful in finding the inverse of a matrix)

*[[Cayley-Hamilton theorem]]

*Stability

*Hermitian Matrices

*Singular values and [[Singular Value Decomposition]]

*Linear differential equations

== Common problems ==

Common problems in linear algebra include:

*simplifying or reducing matrices
*[[Gauss-Jordan elimination]]
*matrix multiplication
*finding inverses and transposes of matrices
*[[Gram-Schmidt procedure]]
*finding [[eigenvalue]]s and [[eigenvector]]s for matrices:
:#find the [[characteristic polynomial]]
:#solve for the eigenvalues
:#solve for the eigenvectors
*finding the orthogonal projection of a vector in a vector space
*diagonalize a matrix
*find the geometric equivalent of a matrix
*finding the determinant of a 2x2 matrix (easy) and a 3x3 matrix (hard)
*finding the inverse of a matrix
*[[decomposition]] or [[factorization]] of a matrix: representing a given matrix as a product of simpler matrices
*QR factorization
*find the least squares solution for a set of data
*use known characteristics of symmetric or diagonalizable matrices to find solutions

== Liberal agenda in linear algebra ==

The study of linear algebra has long had a huge liberal bias, beginning from two of its main constituents, [[Carl Friedrich Gauss]] and [[Wilhelm Jordan]], who were both unfaithful rationalists. From the established terminology of linear algebra, one can see liberals attempting to "rewrite" the classical mathematical terms to suit their radical tastes.

The idea of a [[determinant]] attempts to associate all of the entries of a matrix with a single value. This is much like what liberal politicians such as [[Barack Obama]] are attempting to do to the American public, by generalizing every free American into a single value to be manipulated. Even looking at the word, a "determinant", is reminiscent of [[Hitler]]'s fascist ideas on "determining" the freedom of the German people with [[Nazism]].

Even worse is the idea that a determinant can lead to a [[characteristic polynomial]]. A characteristic polynomial helps liberals to create propaganda about a patriotic matrix known as [[eigenvalues]] and [[eigenvectors]]. Eigenvectors are liberal ideas that can be used to create skewed [[mappings]] and decompositions of matrices. This is very similar to what liberals are attempting to do to the American public, by skewing the news and decomposing American values.

== Application: Analyzing [[Liberal style]] on [[Wikipedia]] ==

An ''n x m'' matrix can be developed using observed incidents of [[liberal style]] in Wikipedia entries, and that data can then be simplified to draw conclusions about how liberal style can mislead viewers. The ''n'' rows can represent different elements of liberal style, while the ''m'' columns can represent different types of entries on Wikipedia.

The prevalence of certain types of liberal style may reflect their perceived effectiveness, and a linear algebra-based approach at modeling it may be useful in debunking the flawed reasoning and claims.

==References==

{{reflist}}

[[Category:Linear algebra]]
[[category:mathematics]]

Linear algebra

2011-11-04T14:11:26Z

PatrioticMath:

'''Linear algebra''' is the branch of [[mathematics]] that deals with [[vector]]s, vector spaces, linear maps, and systems of linear [[equation]]s. Topics studied in Linear algebra include Linear equations, Matrices, [[Matrix]] decompositions, Computations, Vectors, Vector spaces, Multilinear algebra, Affine space, Projective space.<ref>[http://www.m-w.com/cgi-bin/dictionary?book=Dictionary&va=linear+algebra Merriam Webster definition of Linear Algebra]</ref>
Linear algebra has numerous applications in [[engineering]], [[chemistry]] and [[physics]]. Matrices can be used for data fitting tasks such as "curving" student grades on exams, and for finding correlations between two sets of data. Linear algebra is particularly useful for organizing and simplifying data.

The three most vexing computations in linear algebra are these:
#linear equations
#least squares ([[data fitting]])
#finding the eigenvalues of an ''n x n'' matrix for n > 3.

== Basic Concepts ==

*[[Linear equations]]
:*coefficient matrices and Gauss-Jordan elimination
:*[[rank]] (row rank = column rank)
:*geometric representations, especially [[vector]]s and systems with more variables than equations

*[[Linear transformations]]
:*transformations, inverses and matrix products

*Subspaces
:*[[image (mathematics)|image]] and [[kernel (geometry)|kernel]]
:*[[basis]] and [[span]]
:*[[dimension]]

*[[Determinant]]s
:*[[Sarrus's rule]]
:*geometrical interpretation
::*classical adjoint, expansion factor, application to [[parallelepiped]]s)
:*determinant of similar matrix, [[inverse matrix]], product of matrices
:*Cramer's rule (with and without product rule)
:*[[minor]] of a matrix
:*Laplace expansion (cofactors)

*[[Basis]]
:*[[eigenbasis]]
:*[[orthogonal]] eigenbasis
:*[[orthonormal]] eigenbasis

*[[Eigenvalue]]s and [[eigenvector]]s
:*diagonalization
:*[[trace]]
:*[[characteristic polynomial]]
:*complex eigenvalues

*[[Orthogonality]]
:*orthogonal (perpendicular) vectors
:*orthonormal vectors
:*orthogonal projections
:*[[orthogonal matrix]]
:*projections
:*Gram-Schmidt Process and [[QR factorization]]
:*orthogonal matrices, orthogonal transformations
:*data fitting, especially least squares

*[[Symmetric matrix]]
:*[[spectral theorem]]

*[[Quadratic forms]]

*Linear dynamical systems
:*Euler's Formula

=== Notation ===

*matrices are commonly represented by ''A'' and ''B''
*diagonal matrices are represented by ''D''
*an upper triangular matrix is represented by ''R'', as in QR factorization
*when ''A'' is similar to ''B'', then an [[invertible matrix]] ''S'' is used to represent that ''AS'' = ''SB''

== More advanced topics include ==

*[[Vector space]]s (linear spaces)
:*the conditions of a vector space
:*isomorphisms
:*Nth dimensional spaces and subspaces

*Inner spaces
:*inner product spaces

*Determinants
:*cofactor
:*adjugate (useful in finding the inverse of a matrix)

*[[Cayley-Hamilton theorem]]

*Stability

*Hermitian Matrices

*Singular values and [[Singular Value Decomposition]]

*Linear differential equations

== Common problems ==

Common problems in linear algebra include:

*simplifying or reducing matrices
*[[Gauss-Jordan elimination]]
*matrix multiplication
*finding inverses and transposes of matrices
*[[Gram-Schmidt procedure]]
*finding [[eigenvalue]]s and [[eigenvector]]s for matrices:
:#find the [[characteristic polynomial]]
:#solve for the eigenvalues
:#solve for the eigenvectors
*finding the orthogonal projection of a vector in a vector space
*diagonalize a matrix
*find the geometric equivalent of a matrix
*finding the determinant of a 2x2 matrix (easy) and a 3x3 matrix (hard)
*finding the inverse of a matrix
*[[decomposition]] or [[factorization]] of a matrix: representing a given matrix as a product of simpler matrices
*QR factorization
*find the least squares solution for a set of data
*use known characteristics of symmetric or diagonalizable matrices to find solutions

== Liberal agenda in linear algebra ==

The study of linear algebra has long had a huge liberal bias, beginning from two of its main constituents, [[Carl Friedrich Gauss]] and [[Wilhelm Jordan]], who were both unfaithful rationalists. From the established terminology of linear algebra, one can see liberals attempting to "rewrite" the classical mathematical terms to suit their radical tastes.

The idea of a [[determinant]] attempts to associate all of the entries of a matrix with a single value. This is much like what liberal politicians such as [[Barack Obama]] are attempting to do to the American public, by generalizing every free American into a single value to be manipulated. Even looking at the word, a "determinant", is reminiscent of [Hitler]'s fascist ideas on "determining" the freedom of the German people with [Nazism].

Even worse is the idea that a determinant can lead to a [[characteristic polynomial]]. A characteristic polynomial helps liberals to create propaganda about a patriotic matrix known as [[eigenvalues]] and [[eigenvectors]]. Eigenvectors are liberal ideas that can be used to create skewed [[mappings]] and decompositions of matrices. This is very similar to what liberals are attempting to do to the American public, by skewing the news and decomposing American values.

== Application: Analyzing [[Liberal style]] on [[Wikipedia]] ==

An ''n x m'' matrix can be developed using observed incidents of [[liberal style]] in Wikipedia entries, and that data can then be simplified to draw conclusions about how liberal style can mislead viewers. The ''n'' rows can represent different elements of liberal style, while the ''m'' columns can represent different types of entries on Wikipedia.

The prevalence of certain types of liberal style may reflect their perceived effectiveness, and a linear algebra-based approach at modeling it may be useful in debunking the flawed reasoning and claims.

==References==

{{reflist}}

[[Category:Linear algebra]]
[[category:mathematics]]