How to Math

VECTOR SPACES AND SUBSPACES   INTRODUCTION          The definition of a vector space V , whose elements are called vectors , involves arbitrary filed K , whose elements are called scalars .          The following notation will be used (unless otherwise stated or implied)   V  given vectors space u,v,w vectors in  V K  given number field a , b , c , or k scalars in K VECTOR SPACES                   The following define the notion of a vector space V , where K  is the field of scalars. Definition: Let V   be a nonempty set with two operations: 1. Rule of vector addition , which assigns to any u,v ∈ V    a sum u+v  in V . 2. Rule of vector multiplication , which assigns to any u ∈ V  and     k ∈ K  a product ku ∈ K . Then V   is called a vector space (over the field K )   if the following axioms hold:           [A1]       For any vectors  u,v,w ∈ V,   ( u+v ) +w = u+ ( v+w ).           [A2]    There is a vector in  V,  denoted by 𝟢 and called the   zero vector ,

Determinants Introduction         Each n -square matrix A = [a ij ] is assigned a special scalar called the determinant of A , denoted by det ( A )  or | A | or                          An n ✕ n   array of scalars enclosed by straight lines, called a  determinant of order n, is not a matrix but denotes the determinant of the enclosed array of scalars, i.e., the enclosed matrix.           The determinant function was first discovered during the investigation of systems of linear equations. The determinant is an indispensable tool in investigating and obtaining properties of square matrices.   Determinants of Order 1 and 2 The Determinants of Order 1 and 2 are defined as follows: The determinant of a 1x1  matrix  A = [a 11 ]  is the scalar  a 11  itself, that is,                                                    The determinant of order two may easily remembered by using the following diagram:                                                                           

Calculation of a 3x3 matrix determinant using Triangle's Rule. The Triangle’s rule will be formed with this scheme:                                                 The product of diagonal elements and product of elements in the both vertex of two triangles of the first determinant get the “+” sign and the product of diagonal elements and product of elements in the both vertex of two triangles of the second determinant get the “-” sign. In base of triangle’s rule, we have: This method is somehow similar to  Sarrus Rule .

Calculation of 3 ✖ 3 matrix determinant by  Co-factor Expansion along the Column The Co-Factor Expansion Definition : If  A   is a square matrix then the minor of  A ij , denoted by  M ij , which is the determinant of the sub-matrix that results from removing (deleting) the row  ith   and  jth  column of   A . Sign Chart for a 3×3 determinant                                                      A sign chart is either a + or - for each element in the matrix. The first element (row 1, column 1) is always a + and it alternates from there. Note:  The + does not mean positive and the - negative. The + means the same sign as the minor and the - means the opposite of the minor. Think of it as addition and subtraction rather than positive or negative. About the Method To calculate a determinant you need to do the following steps:   1. Pick any column of the matrix 2. Find all the cofactors for that column 3. Multiply each cofactor by its matrix entry 4. Add all the values you've gotten.