Vector Spaces, II
Two, Three, ... N, ... Infinity

One of the recurring themes of mathematics is that the same mathematical ideas can be used to describe and work with many different phenomena. In particular, the algebraic machinery -- vector spaces -- that we are developing based on geometric ideas in R² and R³ can be used in many other, more exotic, situations. We will be able to use geometric thinking in situations that, at first, appear to be very far removed from the usual plane and the usual three-dimensional world. We begin with a short list of some other vector spaces.

A bundle of 5 different things -- for example, apple juice, orange juice, tomato juice, grape juice, and cranberry juice can be represented by a 5-tuple
(x₁, x₂, x₃, x₄, x₅),
or a vector in R⁵.
A bundle of n different things can be represented by an n-tuple
(x₁, x₂, x₃, x₄, .. x_n),
or a vector in Rⁿ.
Sound recordings.
- Until recently sound was recorded using analog equipment. One standard method used magnetic tape with sound represented by the strength of a magnetic field. Mathematically, this can be represented by a function f with f(t) representing the strength of the field that was recorded at time t.
- Currently sound is often recorded digitally -- using a long vector
  (x₁, x₂, x₃, x₄, ... x_n)
  with 44,100 entries for each second of sound and the i-th entry representing the sound at time t = i/44,100 seconds after the start of the recording.
Visual images like television images and the image you are currently looking at on this monitor are often recorded, manipulated, and displayed digitally. For example, the black-and-white photograph below, taken at Capital Reef National Park, is made up of 267 rows, each containing 177 points or pixels. The intensity of the light in each pixel is represented by a real number and the entire image is represented by a matrix with 267 rows and 177 columns. Each entry in the matrix represents the brightness of one pixel. The color version of the same photograph is represented by three matrices -- one for the color red, one for the color green, and one for the color blue.

Each of these examples is a vector space because it has two operations -- vector addition, and scalar multiplication -- that satisfy the usual vector space properties. For example, in Rⁿ vector addition and scalar multiplication are defined by --

Vector addition and scalar multiplication for matrices are defined in the mathematical infrastructure section on matrices. When a matrix represents a black-and-white image then these mathematical operations on matrices correspond to physical operations on images. Vector addition corresponds to superimposing two images and scalar multiplication corresponds to adjusting the brightness. The same thing is true for a color image except that we use three matrices for each image. These ideas are the key to digital processing of images.

In the last module we established a connection between algebraic expressions of the form

P = (1 - t) a + t b

and the geometric idea of points on the line determined by the two vectors a and b. Recall that if t = 0 then P is a; if t = 1 then P is b; and as t moves from 0 to 1, P moves from a to b.

The Java applet below shows how this same algebraic idea is used in movies and television to make a transition from one scene to another. This applet shows three pictures. On the left you see an outside view of the Lincoln Memorial. On the right you see the statue of Abraham Lincon inside the memorial. The picture of the memorial is represented inside your computer as described above by the vector a and the picture of the statue is represented inside your computer by the vector b. The picture in the middle is represented by the vector

P = (1 - t) a + t b

When the applet first starts t = 0 and the middle picture is the same as the picture on the left. You can change the value of t by clicking on the bar at the bottom of the applet. For example click at the far right of the bar. It may take a while for the applet to process your change. You have changed t to 1 and after processing this change the middle picture will show the statue. Now click in the middle of the bar. Now the middle picture will show a picture that is halfway between the two pictures. Try different values of t to see the effect on the middle picture.

If you look at the color picture above from Capital Reef National Park you will notice that the person's face is in the shadow. The image below was adjusted mathematically to change the brightness of a small part of the image. We used scalar multiplication on part of the three matrices representing the image.

Unfortunately the effects of our tampering with the image are quite apparent. Notice the sharp border separating the lightened portion of the image from the remainder of the image. We can use another vector space tool -- moving smoothly from one vector x to another vector y by

P = (1 - t) x + t y

We used this idea above with the Lincoln Memorial and its statue of Lincoln. Now we apply the same idea to part of our usual image -- near the sharp border in the image above we mix the two images using 50% of each, as we move into the area being lightened we use more of the lightened image and as we move away from the area being lightened we use more of the original image. The result is shown below. Notice that we have lightened the person's face without the telltale sharp border. In fact, the evidence of our tampering is barely detectable even when the two photos are examined side-by-side.

Now you can see why people are both delighted and worried about the possibilities for computer manipulatiions of images. Vector space ideas allowed the producers of the movie Forrest Gump to put the title character into a scene with President Kennedy.

Your CAS window contains some experiments using matrices to represent images and the vector space operations on matrices to represent physical manipulations of images. Because the capabilities of the three CAS systems vary so much, the three windows are quite different. Do the experiments described in your CAS window. If you have access to Mathematica we recommend that CAS for this module.

Just as vector space operations can be used to represent physical manipulations of images, they can be used to represent physical manipulations of sound. Multiplying the vector representing a sound by a scalar corresponds to adjusting the volume and vector addition of two sounds corresponds to playing the two sounds at the same time.

Suppose that you are recording a 100 second concert with two intruments -- a violin and a piano. We will represent the sound produced by the piano digitally by the vector P in R^4,410,000 and the sound produced by the violin by the vector V also in R^4,410,000.

You use two microphones -- one placed very close to the piano and one placed very close to the violin. Ideally the first microphone would record only the piano and the second would record only the violin. But, of course, each microphone picks up a little bit of the other instrument. We will represent the recording made by the first microphone by the vector A and the recording made by the second microphone by the vector B. Suppose that

A = 1.0 P + 0.1 V
B = 0.2 P + 1.0 V

How would you process the two recordings digitally using vector space operations to recover the piano playing by itself? answer.
How would you process the two recordings digitally using vector space operations to recover the violin playing by itself?

How would you process the two recordings digitally using vector space operations to produce a recording with both the piano and the violin playing at the same level.?

The applet on the right illustrates some of the power of vector spaces.
Notice the large white triangle. This applet uses the correspondence between colors and geometry produced by vector spaces to let you choose different colors for the triangle. You can place a dot anyplace inside the triangle by moving your mouse and pressing the mouse button. Notice when you move the dot the color of the triangle changes.
We saw in the last module that each point in the triangle can be represented in the form
P = a A + b B + c C
where
a + b + c = 1
and the three coefficients a, b, and c are between zero and one.

Most computers and operating systems represent colors by vectors of the form (R, G, B) in which the three components represent the intensity of red, green, and blue, respectively. None of these components can be negative and one usually represents the maximum intensity for each color on the monitor. Thus, R, G, and B are all between zero and one. The colors of the three vertices are represented by the vectors

R = (1, 0, 0)
G = (0, 1, 0)
B = (0, 0, 1)

When you place the dot at the point represented by the vector

P = a A + b B + c C

where A represents the red vertex; B represents the green vertex; and B represents the blue vertex and

a + b + c = 1

and the three coefficients a, b, and c are between zero and one the triangle is colored using the color

a R + b G + c B

Vector Spaces, II Two, Three, ... N, ... Infinity

Vector Spaces, II
Two, Three, ... N, ... Infinity