Lecture 5: Vectors, dot products, solutions of Ax=b (Nicholson, Sections 4.1 & 4.2)

Alternate Video Access via MyMedia | Video Duration: 27:45 |

Description: Missing lecture: The next thing in Nicholson’s book is Section 1.3 “homogeneous systems of linear equations; trivial and non-trivial solutions; linear combinations of solutions; basic solutions”. In the previous book and in the lectures from Fall 2016 and Fall 2017, these concepts were introduced but were interwoven with other material which you haven’t been introduced to yet. There’s no way to disentangle the material and so I have no lecture videos to offer you on the topic. Started with a quick review of Cartesian coordinates, vectors, vector addition, and scalar multiplication of vectors. The most important thing to keep track of is the difference between a point P(p1,p2) which has coordinates p1 and p2 and the position vector of this point P which is a vector whose tail is at the origin and whose tip is at P(p1,p2). In the course, we often use position vectors and points interchangeably and this can be very confusing sometimes.

5:58 – 6:22 --- ignore this part.

6:22 --- introduced the the dot product for vectors in R^2. First defined algebraically: the dot product of u = [u1;u2] and v = [v1;v2] is u1*v1+u2*v2. Second, defined it geometrically: if you know the length of the two vectors and the angle between them then you can construct the dot product. (This leads to the natural question: if I gave a pair of vectors to two students and asked student A to compute the dot product using the algebraic definition and asked student B to compute the dot product using the geometric definition, would students A and B always get the exact same answer?) The dot product can be defined in two ways and the two different ways of defining it give the same answer and are useful in different ways. This is a powerful and important aspect of the dot product, not discussed in the book.

10:00 --- generalized the dot product and length to vectors in R^n.

12:10 --- introduced what it means for two vectors to be orthogonal. Note! The zero vector is orthogonal to all vectors.

19:40 --- stated the theorem that tells us how the dot product interacts with scalar multiplication and with vector addition.

23:30 Multiplying a matrix and a vector to verify a solution of a system of linear equations.