Linear algebra

Linear Algebra History And Application: Linear algebra is the branch of mathematics that deals with the study of vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. The term ‘linear’ means a one dimensional set of rules and regulations through which a system achieves its equilibrium state. Linear algebra applications are in abstract three dimensional algebra, and functional analysis of electrical instruments.

The present day linear algebra originated 2,000 years ago in China. Chinese scholars first used the Nine Chapters on the Mathematical Art (Jiuzhang Suanshu) as the basics of one dimensional theory. Chinese mathemeticians formulated a system, in which they organized linear equations in a rectangular pattern, known as Fang Cheng, involving horizontal and vertical rods. This rectangular representation of linear equations is the same one used in the modern day matrix. Present day linear algebra has developed a more modern approach known as the Gauss-Jordan matrix elimination.

Graphing A Linear Equation: Consider an algebraic equation, such as y = 2x + 7 or 3x + 2y - z = 4. In the equation, the highest degree term in the variable or variables is the first-degree term. Such an equation is considered linear. The graph of this equation is a straight line, if there are two variables. Algebraically the equation is y = mx + b.

Graphing of a linear equation

• The Y intercept is located on the graph and the subsequent point is plotted
• From this point, using the slope which is the coefficient of x, the x intercept is derived
• The line is then drawn on the graph with a proper scale

Linear equations can be solved through linear algebra. The coefficients of linear equations form the rows and columns of matrices, with the variables forming an adjacent matrix. The equation is then solved using Clayey – Hamilton’s theorem, which can work out the coefficients up to the ‘n’th degree.

Linear equations are applicable in calibrating and measuring various modes of electricity in a network.

Linear Algebra Axler Contribution: Sheldon Axler is Dean of the College of Science & Engineering at San Francisco State University. He received the Lester Ford Award for writing for the Mathematical Association of America. His books Linear Algebra Done Right and Harmonic Functions are used as textbooks by many universities and colleges.

Sheldon Axler was Associate Editor of the American Mathematical Monthly and the Editor-in-Chief of the Mathematical Intelligencer. He is on the board for a three book series, Graduate Texts in Mathematics, Undergraduate Texts in Mathematics, and Universitext.

Sheldon Axler has developed the factorization theory $L^/infinity functions.
This theory gives a generalised factorization theory for laplace function’s L operators raised to power of n. His lecture note in facorization is about using the theorem on the Blasche product, in which he states “consider g to be an element of L infinity. Then there exists a Blasche product b and a function h member of the h infinity such that g = h/b”

The lecture is for elementary undergraduates and is available as an online course.