Assigned Monday 7/16:
Read sections 2.5 and 8.1.
Complete the following problems:
6.7: 17
2.2: 24
2.3: 10, 19, 20, 29, 31, 35
2.4: 1
2.5: 3, 8
Essay question due 7/23:
Graphing vectors in the xy - coordinate plane:
Column matrices such as,
|1|
|0|
are also called column vectors. (Denote this particular column vector by v.) To draw a 2x1 column vector in the xy - coordinate plane view the top entry as the x coordinate and the bottom entry as the y coordinate. Plot the resulting point (so the point (1,0) in the case of v), and then draw an arrow from the origin to that point.
The graph of v is:
The graph of the column vector,
| 2|
|-3|
is below:
In this problem we are going to look at the effect of matrix multiplication on vectors in the plane. Let's fix our `point of reference vector' to be the vector v from above.
Let A be the matrix below:
|cos(t) sin(t)|
|-sin(t) cos(t)|
Also let B be the matrix:
|s 0|
|0 s|
(A) Set s equal to 2 in matrix B. Now draw v in the plane, and then draw Bv in the plane. How does multiplication by B, with s equal to 2, effect v?
(B) Explain in general how multiplication by B effects v. (That is, don't specify s. Explain the effect for an arbitrary number s.)
(C) Set t equal to 45 degrees in A. Now draw v in the plane, and then draw Av in the plane. How does multiplication by A, with t equal to 45 degrees, effect v?
(D) Explain in general how multiplication by A effects v. (Again, don't specify t. Explain the effect for an arbitrary angle measure t.)
(E) Draw the vector,
|-3|
|3|
in the plane. What matrices do you need to multiply this vector by in order to move it back to v? How does this relate to the inverse matrix of A and the inverse matrix of B?
Connection to computer animation:
Pixels on a video screen can be described by their x and y coordinates. Thus an image on a video screen is just an assignment of colors to the different points (x,y) on the screen. To move an image around we just multiply all of the points on the screen by a matrix such as A or B.
COURSE INFO