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MatricesAndStems - Today we are going to move from vectors...

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Copyright © Paul Kantor, 2011. Today we are going to move from vectors to some concepts that are very important for searching and other things that computers can do with texts. These things are called stems or roots of words, and they are the first step towards having computers ‘recogni ze concepts’ in texts. (0) A note about the “cheat sheet” that you can take to exams. It should always include the tables of logarithms and trigonometric functions. And when you copy formulas into it, be very careful to copy themcorrectly, or they will not be much use. Now for the material. 1 Dimensions, Components and Bases. . Let’s recall that a vector is an indexed array of numbers. For discussion, we have to write them in some orcer, and so, on the blackboard, or kin the notes, a vector is actually an ordered set of numbers. For example, a=(1,3,5) is a vector, and so is b=(72, -3, -4, 6). The individual numebr are called the components of the vectors.. So, for example, “3” is a component of the first vector, and 3 is a component of the second vector. In writing about them we use some subscript (usually i or j or k ) to represent the position that the component occupies. So we would write that 2 3 a or 2 3, b   and so forth, Clearly the number of components that the vector has is important. This number is called the dimensionality or the dimension of the vector. It is sometimes written as Dim(v) where v is the name of the vector. So Dim(a)=3; Dim(b)=4. What is the dimensionality Dim(c), when c= (2,3,0,5,0,4)? The answer is 6 (not 4) because the zeroes are numbers, and they count. Addition and dimension. We recall that we can add vectors by adding the corresponding componenets. For example, the sum of a=(1,2,3) and b=(3,2,1) is just (4,4,4). But lets go back to our example of , a=(1,3,5) is a vector, and so is b=(72, -3, -4, 6). Can we add them? The answer is NO !. They have different numbers of components, and so adding them is not even defined. Sometimes people think that we could pad a by putting another 0 in it. But that is simply not allowed (we would not know whether to put it first, or last, or somewhere in the middle, if we tried.). 1.1 What do the components mean? As pure mathematicians, we can happily manipulate vectors, without worrying about what they meamn. That is one of the great powers of mathematics as a tool. But the fact that they do meansomehting, in particular cases, tells us to be careful about how we use them.
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Copyright © Paul Kantor, 2011. When the mathematics of vectors is applied to any particular situation in the real world (like motions) or in the real world of abstractions (like searching), the components are defined with respect to some more fundamental ntions, called the basis of the vectors. 1.1.1 Example: A basis in physical space We have looked at vectors that represent motion, East, or Up, or North. So we might have the vector (2,1,0) which means (2 steps East; 1 step up; 0 steps North).
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