A Review of Concepts Concerning Linear Algebra and the Applications of Similarity Transformations

Linear algebra is a field of mathematics which concerns itself with the properties and operations of linear spaces and linear transformations which are functions mapping from one linear space to another based on a set of axioms they must follow. Linear transformations are represented via matrices. While linear spaces and the elements that constitute a linear space are defined formally in terms of axioms/properties concerning the addition, multiplication and other operations on the elements [5], it is best to start linear algebra with Euclidean spaces and vectors that constitute the Euclidean space. The 3 dimensional Euclidean space and the Cartesian geometry which we do in that space provide good intuitions for understanding linear algebra. These can be generalized to n dimensions. Concepts such as lines, planes, parallelism, projections and orthogonality have neat formulations and generalizations in linear algebra. We start with vectors to represent points in 3 D space, and matrices as mathematical objects which stretch and rotate these vectors. We see that this is related to 3 linear equations with 3 unknowns. We then review and formalize concepts such as linear spaces, basis vectors, independence, projections, orthogonality, diagonalization of a matrix etc. Linear algebra is deeply connected with multivariable calculus and optimization; concepts such as the derivative of a vector field and multivariable Taylor Series require the understanding of linear algebra. Today, linear algebra is getting even more important, from principal component analysis in machine learning to Fourier analysis in digital signal processing to the Lorentz transformation in Einstein's Relativity, all are based on the concepts of linear transformations and change of basis. Citation: Advait Pravin Savant. "A Review of Concepts Concerning Linear Algebra and the Applications of Similarity Transformations." Global Research and Development Journal For Engineering 5.12 (2020): 1 - 7.