Mathematics 241 – Multivariable Calculus Discussion 1 – Extending Vector Operations to R^{n} September 7, 2007

Background

The definitions of R^{2} and R^{3} can be extended to give n-dimensional spaces R^{n} for any n ≥ 1:

R^{n} = {(a_{1}, …, a_{n}) : a_{i} ∈ R for all i = 1, …, n}.

We will see today that many of the vector operations we have considered in 2 and 3 dimensions can be extended to these spaces as well. But first we want to consider a question that might be at the forefront of your minds! Why take n > 3?

Discussion Questions

A) R^{2} has the obvious interpretation of the coordinate plane from geometry. R^{3} can be seen as a mathematical model of the (apparently) 3-dimensional physical world we live in. Although the mathematical definition of R^{4} or R^{n} more generally presents little difficulty, it is still natural to ask why might the mathematical object R^{n} for n ≥ 4 be useful? Here are some examples:

1) In physics, when considering objects in motion, we often need to know not just where an object is, but when it is there. If we think of the “complete coordinates” of an object as its three spatial coordinates, and the current time, what space R^{n} contains all the possible “complete coordinate vectors?”

2) Again in physics, suppose we are interested in the configurations of several objects moving in space simultaneously. For instance, say we have the configuration of the center of the Sun, the center of the Earth, and the center of the Moon. To describe each center’s position, we need three spatial coordinates, so how many coordinates do we need to specify the whole configuration (Sun, Earth, Moon)? What space R^{n} contains all the possible “configuration vectors?”

3) In economics, suppose we wanted to represent the current state of the world economy by giving the GNP’s of each of the 190 or so countries in the world. What space R^{n} contains all the possible “world economy state vectors?”

All of these are realistic examples of the way spaces R^{n} for n ≠ 2, 3 are used to study situations mathematically(!)

B) A vector sum operation can be defined in R^{n} for any n:

(a_{1}, …, a_{n}) + (b_{1}, …, b_{n}) = (a_{1} + b_{1}, …, a_{n} + b_{n}).

Each vector in R^{n} can be written in one and only one way as a linear combination of a collection of vectors analogous to the standard basis i, j, k in R^{3}. What are these vectors, and how does this work?

C) A dot product can be defined for vectors in R^{n} by

a · b = (a_{1}, …, a_{n}) · (b_{1}, …, b_{n}) = a_{1}b_{1} + ⋯ + a_{n}b_{n}.

1) Show that a · a ≥ 0 for all a in R^{n}, and a · a = 0 only if a = (0, …, 0).

2) Show that a · b = b · a for all a, b in R^{n}.

3) Show that

a · (b + c) = a · b + a · c

for all a, b, c in R^{n}.

4) Say how to define a norm ∥a∥ for vectors a ∈ R^{n}.

5) Show that the Cauchy-Schwarz inequality:

|a · b| ≤ ∥a∥∥b∥

holds for all a, b in R^{n}. (Hint: it suffices to show

(a · b)^{2} ≤ ∥a∥^{2}∥b∥^{2};

do you see why?)