《机器学习与知识发现》教学资源:Linear Algebra Review and Reference
1 Basic Concepts and Notation Linear algebra provides a way of compactly representing and operating on sets of linear equations.For example,consider the following system of equations: 4x1-5x2=-13 -2x1+3x2=9. This is two equations and two variables,so as you know from high school algebra,you can find a unique solution for x and z2(unless the equations are somehow degenerate,for example if the second equation is simply a multiple of the first,but in the case above there is in fact a unique solution).In matrix notation,we can write the system more compactly as Ax=b with 4=[4],=[] As we will see shortly,there are many advantages (including the obvious space savings) to analyzing linear equations in this form. 1.1 Basic Notation We use the following notation: By A E Rmxn we denote a matrix with m rows and n columns,where the entries of A are real numbers. By z E R",we denote a vector with n entries.By convention,an n-dimensional vector is often thought of as a matrix with n rows and 1 column,known as a column vector. If we want to explicitly represent a row vector-a matrix with 1 row and n columns -we typically write zT(here aT denotes the transpose of x,which we will define shortly). The ith element of a vector x is denoted xi: T1 T= In 21 Basic Concepts and Notation Linear algebra provides a way of compactly representing and operating on sets of linear equations. For example, consider the following system of equations: 4x1 − 5x2 = −13 −2x1 + 3x2 = 9. This is two equations and two variables, so as you know from high school algebra, you can find a unique solution for x1 and x2 (unless the equations are somehow degenerate, for example if the second equation is simply a multiple of the first, but in the case above there is in fact a unique solution). In matrix notation, we can write the system more compactly as Ax = b with A =  4 −5 −2 3  , b =  −13 9  . As we will see shortly, there are many advantages (including the obvious space savings) to analyzing linear equations in this form. 1.1 Basic Notation We use the following notation: • By A ∈ R m×n we denote a matrix with m rows and n columns, where the entries of A are real numbers. • By x ∈ R n , we denote a vector with n entries. By convention, an n-dimensional vector is often thought of as a matrix with n rows and 1 column, known as a column vector. If we want to explicitly represent a row vector — a matrix with 1 row and n columns — we typically write x T (here x T denotes the transpose of x, which we will define shortly). • The ith element of a vector x is denoted xi : x =      x1 x2 . . . xn      . 2
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