## Knowledge in Metrics Matrices, System of Linear Equations, Gauss Elimination Method

n linear algebra, Gaussian elimination (also known as row reduction) is an algorithm for solving systems of linear equations. It is usually understood as a sequence of operations performed on the corresponding matrix of coefficients. Echelon Form, Elementary Matrices & Row Reduction

Elementary Operations. There are three kinds of elementary matrix operations. Multiply each element in a row (or column) by a non-zero number. Multiply a row (or column) by a non-zero number and add the result to another row (or column). Determinant and its properties

Properties of determinants Interchange two rows or cols changes the sign: -> -1 * det(A) ... transpose -> det (A) unchanged. ... multiply row * k -> k * det(A) ... multiply matrix * k -> k^2 * det(A) ... det (A B) -> det(A) * det(B) ... proportional rows or columns -> det() == 0. ... Add multiple of one row to another -> det unchanged. ... Geometric interpretation. Dimension of Vector Space

Image result for : Dimension of Vector Spacemath.stackexchange.com In mathematics, the dimension of a vector space V is the cardinality (i.e. the number of vectors) of a basis of V over its base field. It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. Linear Transformation & Rank-Nullity Theorem

Image result for Linear Transformation & Rank-Nullity The rank-nullity theorem is a fundamental theorem in linear algebra which relates the dimensions of a linear map's kernel and image with the dimension of its domain.