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MIT 18 01 - Vectors and Matrices

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CHAPTER 11 Vectors and Matrices This chapter opens up a new part of calculus. It is multidimensional calculus, because the subject moves into more dimensions. In the first ten chapters, all functions depended on time t or position x-but not both. We had f(t) or y(x). The graphs were curves in a plane. There was one independent variable (x or t) and one dependent variable (y or f). Now we meet functions f(x, t) that depend on both x and t. Their graphs are surfaces instead of curves. This brings us to the calculus of several variables. Start with the surface that represents the function f(x, t) or f(x, y) or f(x, y,,t). I emphasize functions, because that is what calculus is about. EXAMPLE 1 f(x, t) = cos (x -t) is a traveling wave (cosine curve in motion). At t = 0 the curve is f = cos x. At a later time, the curve moves to the right (Figure 11.1). At each t we get a cross-section of the whole x-t surface. For a wave traveling along a string, the height depends on position as well as time. A similar function gives a wave going around a stadium. Each person stands up and sits down. Somehow the wave travels. EXAMPLE 2 f(x, y) = 3x + y + 1 is a sloping roof (fixed in time). The surface is two-dimensional-you can walk around on it. It is flat because 3x + y + 1 is a linear function. In the y direction the surface goes up at 45". If y increases by 1, so does f. That slope is 1. In the x direction the roof is steeper (slope 3). There is a direction in between where the roof is steepest (slope fi). EXAMPLE 3 f(x, y, t) = cos(x -y -t) is an ocean surface with traveling waves. This surface moves. At each time t we have a new x-y surface. There are three variables, x and y for position and t for time. I can't draw the function, it needs four dimensions! The base coordinates are x, y, t and the height is f.The alternative is a movie that shows the x-y surface changing with t. At time t = 0the ocean surface is given by cos (x -y). The waves are in straight lines. The line x -y = 0follows a crest because cos 0= 1. The top of the next wave is on the parallel line x -y = 2n, because cos 2n = 1. Figure 11.1 shows the ocean surface at a fixed time. The line x -y = t gives the crest at time t. The water goes up and down (like people in a stadium). The wave goes to shore, but the water stays in the ocean.11 Vectors and Matrices Fig. 11.1 Moving cosine with a small optical illusion-the darker Fig. 11.2 Linear functions give planes. bands seem to go from top to bottom as you turn. Of course multidimensional calculus is not only for waves. In business, demand is a function of price and date. In engineering, the velocity and temperature depend on position x and time t. Biology deals with many variables at once (and statistics is always looking for linear relations like z = x + 2y). A serious job lies ahead, to carry derivatives and integrals into more dimensions. In a plane, every point is described by two numbers. We measure across by x and up by y. Starting from the origin we reach the point with coordinates (x, y). I want to describe this movement by a vector-the straight line that starts at (0,O) and ends at (x, y). This vector v has a direction, which goes from (0,O) to (x, y) and not the other way. In a picture, the vector is shown by an arrow. In algebra, v is given by its two components. For a column vector, write x above y: v = [,I (x and y are the components of v). Note that v is printed in boldface; its components x and y are in lightface.? The vector -v in the opposite direction changes signs. Adding v to -v gives the zero vector (different from the zero number and also in boldface): X-X and vv=[ -0.]=[:I Y-Y Notice how vector addition or subtraction is done separately on the x's and y's: ?Another way to indicate a vector is 2You will recognize vectors without needing arrows.1 1 .I Vectors and Dot Products Fig. 11.3 Parallelogram for v + w, stretching for 2v, signs reversed for -v. The vector v has components v, = 3 and v, = 1. (I write v, for the first component and v, for the second component. I also write x and y, which is fine for two com- ponents.) The vector w has w, = -1 and w, = 2. To add the vectors, add the com- ponents. To draw this addition, place the start of w at the end of v. Figure 11.3 shows how w starts where v ends. VECTORS WITHOUT COORDINATES In that head-to-tail addition of v + w, we did something new. The vector w was moved away from the origin. Its length and direction were not changed! The new arrow is parallel to the old arrow-only the starting point is different. The vector is the same as before. A vector can be defined without an origin and without x and y axes. The purpose of axes is to give the components-the separate distances x and y. Those numbers are necessary for calculations. But x and y coordinates are not necessary for head- to-tail addition v + w, or for stretching to 2v, or for linear combinations 2v + 3w. Some applications depend on coordinates, others don't. Generally speaking, physics works without axes-it is "coordinate-free." A velocity has direction and magnitude, but it is not tied to a point. A force also has direction and magnitude, but it can act anywhere-not only at the origin. In contrast, a vector that gives the prices of five stocks is not floating in space. Each component has a meaning-there are five axes, and we know when prices are zero. After examples from geometry and physics (no axes), we return to vectors with coordinates. EXAMPLE 1 (Geometry) Take any four-sided figure in space. Connect the midpoints of the four straight sides. Remarkable fact: Those four midpoints lie in the same plane. More than that, they form a parallelogram. Frankly, this is amazing. Figure 1 1.4a cannot do justice to the problem, because it is printed on a flat page. Imagine the vectors A and D coming upward. B and C go down at different angles. Notice how easily we indicate the four sides as vectors, not caring about axes or origin. I will prove that V = W. That shows that the midpoints form a parallelogram. What is V? It starts halfway along A and ends halfway along B. The small triangle at the bottom shows V = $A + 3B. This is vector addition-the tail of 3B is at the head of 4A. Together they equal the shortcut V. For the same reason W = 3C + …


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MIT 18 01 - Vectors and Matrices

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