How to use Linear Combinations

-

LINEAR COMBINATIONS

         Let V be a vector space over a field K. A vector v in V is a linear combination of vectors u₁+u₂+⋯+uₙ in V if there exist scalars a₁,a₂,⋯,aₙ in K such that 

                                                v = a1u1+a₂u₂+⋯+aₙuₙ

Alternatively, v is a linear combination of vectors u₁+u₂+⋯+uₙ  if there is a solution to the vector equation

                                                v = x1u1+x₂u₂+⋯+xₙuₙ

where x₁+x₂+⋯+xₙ 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

u1 = (1, 2, 3), u2 = (2, 3, 7), u3 = (3, 5, 6) 

 

Solution:

         We seek scalars x, y, z such that  v = x1u1+x₂u₂+⋯+xₙuₙ, that is,

                    

            Reducing the system to echelon form yields

            Back-substitution yields the solution

                                                

            Thus,







| BS in Math | MS in Applied Math (30-units earned) | MS in Pure and Applied Math (Thesis still on progress) | Full-time Private Tutor | Online Math Instructor |

Comments

Post a Comment

Popular posts from this blog

How to use Triangle's Rule in Finding Determinant of a 3x3 Matrix

-
Image
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 .
Read more

How to use Co-factor Expansion along the Column in Finding Determinant of a 3x3 Matrix

-
Image
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.  
Read more