Mathematical Structure
The Finite Dimensional Projection Theorem

Prerequisites:

• Groups
• Vector Spaces
• Magnitude or Length
• Dot Product

You should use one of the computer algebra systems below with the following exercises. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.

One of the most important tools we use for analyzing something is breaking it into parts. For example, if a force acts on a moving object we break the force into two components -- one component that is tangent to the motion of the object and another component that is perpendicular to the motion of the object. The tangential component causes the object to speed up or slow down and the perpendicular component causes it to turn.

In this module we develop a series of three very similar theorems that enable us to break a vector into parts.

Theorem

Suppose that x is a vector and that u is a unit vector. Let

Proof:

The first conclusion is obvious. For the second conclusion notice that

1. Suppose that an object is moving in the direction given by the unit vector u = (3/5, 4/5) and that the force F = (1, 2) is acting on the object. Find the component of the force that is parallel to the motion of the object and the component of the force that is perpendicular to the object. answer.

2. Break the vector x = (3, 5) up into two components one of which is parallel to the vector w = (1, 1) and the other of which is perpendicular to w. Warning: Notice that w is not a unit vector. answer.

3. Suppose that x is a vectors and that w is a nonzero vector. Find a general formula for breaking x into two components one of which is parallel to w and the other of which is perpendicular to w. answer.

4. Let C[-1, 1] denote the vector space of continuous functions on the interval [-1, 1] with the usual dot product. Decompose the function f(x) = x^3 into two pieces one of which is parallel to the function g(x) = x and the other of which is perpendicular to g(x). answer.

5. Let C[-pi, pi] denote the vector space of continuous functions on the interval [-pi, pi] with the usual dot product. Decompose the function f(x) = 2 sin(x + 0.25) into two components one of which is a multiple of the function g(x) = sin x and the other of which is perpendicular to g(x). answer.

Theorem:

Suppose that x is a vector and that u and v are perpendicular unit vectors. Let

• determine the appearance of a three dimensional object viewed from a distant point.

• find the frequency w component of a signal or data.

1. Decompose the vector x = (1, 2, 3) into two components one of which is in the plane generated by the vectors

   u = (3/5, 4/5, 0) and v = (-4/5, 3/5, 0)

   and the other of which is perpendicular to this plane. answer .

2. Decompose the vector x = (1, 2, 3) into two components one of which is in the plane generated by the vectors

   s = (1, 1, 0) and t = (-1, 1, 0)

   and the other of which is perpendicular to this plane. Warning: the two vectors s and t are perpendicular but they are not unit vectors. answer .

3. Suppose that u and v are two vectors that are perpendicular but not necessarily unit vectors. Find a general formula for decomposing a vector x into two components one of which is in the plane generated by the vectors u and v and the other of which is perpendicular to this plane. answer .

4. Let C[0, 1] denote the vector space of continuous functions on the interval [0, 1] with the usual dot product. Decompose the function

   f(x) = 3 sin(2 pi x + 0.15) + sin(4 pi x)

   into two components one of which is in the plane generated by the functions

   sin 2 pi x and cos 2 pi x

   and the other of which is perpendicular to this plane. answer .

Theorem:

Suppose that x is a vector and u₁, u₂, ... u_n are perpendicular unit vectors. Let

This theorem is the mathematical tool that we use to

• analyze a periodic function or periodic data