linear Algebra 1

 

Welcome My dear students to the corse of linear algera 1 here at the university of wales,

 Im mr Chadwick Petersen, and I will be your teacher for the Whole semester 

Topic 1

assignment 1

assignment due: Jan/12/2022

Find the course book at this Wales link  and read the first four chapters 

The procedure (using counting rods) for solving simultaneous linear equations now called Gaussian elimination appears in the ancient Chinese mathematical text Chapter Eight: Rectangular Arrays of The Nine Chapters on the Mathematical Art. Its use is illustrated in eighteen problems, with two to five equations.[4]Systems of linear equations arose in Europe with the introduction in 1637 by René Descartes of coordinates in geometry.

Topic 2

 

 In fact, in this new geometry, now called Cartesian geometry, lines and planes are represented by linear equations, and computing their intersections amounts to solving systems of linear equations.The first systematic methods for solving linear systems used determinants, first considered by Leibniz in 1693. In 1750, Gabriel Cramer used them for giving explicit solutions of linear systems, now called Cramer's rule. Later, Gauss further described the method of elimination, which was initially listed as an advancement in geodesy.[5]In 1844 Hermann Grassmann published his "Theory of Extension" which included foundational new topics of what is today called linear algebra.

Topic 3

assignment 3

assignment due: Jan/19/2022

 In 1848, James Joseph Sylvester introduced the term matrix, which is Latin for womb.Linear algebra grew with ideas noted in the complex plane. For instance, two numbers w and z in {\displaystyle \mathbb {C} }\mathbb {C} have a difference w – z, and the line segments {\displaystyle {\overline {wz}}}{\displaystyle {\overline {wz}}} and {\displaystyle {\overline {0(w-z)}}}{\displaystyle {\overline {0(w-z)}}} are of the same length and direction. The segments are equipollent. The four-dimensional system {\displaystyle \mathbb {H} }\mathbb {H} of quaternions was started in 1843. The term vector was introduced as v = x i + y j + z k representing a point in space. The quaternion difference p – q also produces a segment equipollent to {\displaystyle {\overline {pq}}.}{\displaystyle {\overline {pq}}.} Other hypercomplex number systems also used the idea of a linear space with a basis.Arthur Cayley introduced matrix multiplication and the inverse matrix in 1856, making possible the general linear group.

Topic 4

assignment 4

assignment due: Jan/24/2022 at 3:00 PM

 The mechanism of group representation became available for describing complex and hypercomplex numbers. Crucially, Cayley used a single letter to denote a matrix, thus treating a matrix as an aggregate object. He also realized the connection between matrices and determinants, and wrote "There would be many things to say about this theory of matrices which should, it seems to me, precede the theory of determinants".[5]Benjamin Peirce published his Linear Associative Algebra (1872), and his son Charles Sanders Peirce extended the work later.[6]The telegraph required an explanatory system, and the 1873 publication of A Treatise on Electricity and Magnetism instituted a field theory of forces and required differential geometry for expression. Linear algebra is flat differential geometry and serves in tangent spaces to manifolds.

Topic 5

assignment 5

assignment due: Feb/1/2022 at 1:00 PM

 Electromagnetic symmetries of spacetime are expressed by the Lorentz transformations, and much of the history of linear algebra is the history of Lorentz transformations.The first modern and more precise definition of a vector space was introduced by Peano in 1888;[5] by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Linear algebra took its modern form in the first half of the twentieth century, when many ideas and methods of previous centuries were generalized as abstract algebra. The development of computers led to increased research in efficient algorithms for Gaussian elimination and matrix decompositions, and linear algebra became an essential tool for modelling and simulations

Topic 5

assignment 5

Assignment due: Feb/3/2022 at 10:00 AM

You need to talk about number fields 

In mathematics, an algebraic number field is an extension field K of the field of rational numbers \mathbb {Q} such that the field extension K/{\mathbb {Q}} has finite degree. Thus K is a field that contains \mathbb {Q} and has finite dimension when considered as a vector space over

Topic 6

assignment 6

Assignment due: feb/15/2022 at 4:00 PM

tallk about the row space 

The row space The row space of a matrix is the collection of all linear combinations of its rows. Equivalently, the row space is the span of rows. The elements of a row space are row vectors. If a matrix has m columns, its row space is a subspace of (the row version of) Rm.

 

Topic 7

assignment 7

Assignment due: feb/20/2022 at 8:00 AM

tallk about the collumn space 

In linear algebra, the column space (also called the range or image) of a matrix A is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix is the image or range of the corresponding matrix transformation.

Topic 8

assignment 8

Assignment due: feb/20/2022 at 10:00 AM

tallk about the row matrix

Row matrix is a matrix having all its elements in a single row. The elements are arranged in a horizontal manner, and the order of a row matrix is 1 x n. A row matrix, A = [a, b, c, d] has only one row and can have numerous columns, which are equal to the number of elements in the row.

Topic 9

assignment 9

Assignment due: feb/27/2022 at 11:00 PM

tallk about the row matrix

A column matrix is a type of matrix that has only one column. The order of the column matrix is represented by m x 1, thus the rows will have single.

here is a link to a book on matrices

Topic 10

assignment 10

Assignment due: Mar/6/2022 at 10:00 AM

talk about the inverse of a matrix

In linear algebra, an n-by-n square matrix A is called invertible, if there exists an n-by-n square matrix B such that \mathbf {AB} =\mathbf {BA} =\mathbf {I} _{n}\ where Iₙ denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.

here is a link to a manual on inverse matrices