MATH125 Lecture Notes - Lecture 2: Dot Product, Linear Combination, Scalar Multiplication

• School: University of Alberta
• Department: Mathematics
• Course: MATH125
• Professor: Vladimir Chernousov

Document Summary

Remark: because of vector operation theorem 2, u+v+w can be written without parentheses. Example of vector operations question: simplify following: 3a+(5b-2a) + 2(b-a) = 3a + 5b - 2a + 2b - 2a. Then the vector c1v1 +c2v2,,+ckvk is called the linear combination of vectors v1vk with coefficients c1ck. Vector v in r^n is called a linear combination of vectors v1vk if there exists scalars c1ck such that v=c1v1++ckvk. Definition: let u=[u1,u2,,un] and v=[v1,b2,,vn] be vectors in r^n. Then the dot product is denoted by u v=[u1v1+u2v2++unvn] which is a real number scalar quantity. *remark: if u=r^2 and v=r^3 then there is no dot product for these vectors, because there is no third coordinate in the u vector. Therefore vectors must be in the same dimension for a dot product to be made. Example: u=[1,2] v=[3,4] then u v=[1*3+2*4] = [3+8] = 11.