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LINEAR COMBINATIONS          Let V  be a vector space over a field K . A vector v  in V  is a linear combination of vectors  u ₁ + u ₂ + ⋯+ u ₙ  in V  if there exist scalars a ₁, a ₂ , ⋯, a ₙ  in  K   such that                                                             v =  a 1 u 1 + a ₂ u ₂ + ⋯+ a ₙ u ₙ Alternatively, v   is a linear combination of vectors  u ₁ + u ₂ + ⋯+ u ₙ     if there is a solution to the vector equation                                                 v =  x 1 u 1 + x ₂ u ₂ + ⋯+ x ₙ u ₙ where x ₁ + x ₂ + ⋯+ x ₙ   are unknown scalars.   Linear Combinations in  Rⁿ          The question of expressing one vector in  Rⁿ  as linear combination of other vector in  Rⁿ   is equivalent to solving a nonhomogeneous system of linear equations, as illustrated below. a)   Suppose we want to express v = (3, 7, -4)  as a linear combination of the vectors u 1  =  (1, 2, 3),  u 2  =  (2, 3, 7),  u 3  =  (3, 5, 6)    Solution:          We seek scalars x ,

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: