Ch4_Linear_Algebra - Chapter 4 Vectors Matrices and Linear...

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Chapter 4: Vectors, Matrices, and Linear Algebra Scott Owen & Greg Corrado Linear Algebra is strikingly similar to the algebra you learned in high school, except that in the place of ordinary single numbers, it deals with vectors. Many of the same algebraic operations you’re used to performing on ordinary numbers (a.k.a. scalars), such as addition, subtraction and multiplication, can be generalized to be performed on vectors. We’ll better start by defining what we mean by scalars and vectors . Definition: A scalar is a number. Examples of scalars are temperature, distance, speed, or mass – all quantities that have a magnitude but no “direction”, other than perhaps positive or negative. Okay, so scalars are what you’re used to. In fact we could go so far as to describe the algebra you learned in grade school as scalar algebra, or the calculus many of us learned in high school as scalar calculus, be cause both dealt almost exclusively with scalars. This is to be contrasted with vector calculus or vector algebra, that most of us either only got in college if at all. So what is a vector? Definition: A vector is a list of numbers . There are (at least) two ways to interpret what this list of numbers mean: One way to think of the vector as being a point in a space . Then this list of numbers is a way of identifying that point in space, where each number represents the vector’s component that dimension. Another way to think of a vector is a magnitude and a direction , e.g. a quantity like velocity (“the fighter jet’s velocity is 250 mph north-by-northwest”). In this way of think of it, a vector is a directed arrow pointing from the origin to the end point given by the list of numbers. In this class we’ll denote vectors by including a small arrow overtop of the symbol like so: ! a . Another common convention you might encounter in other texts and papers is to denote vectors by use of a boldface font ( . An example of a vector is ! a = [4, 3] . Graphically, you can think of this vector as an arrow in the x-y plane, pointing from the origin to the point at x =3, y =4 (see illustration.) In this example, the list of numbers was only two elements long, but in principle it could be any length. The dimensionality of a vector is the length of the list. So, our example ! a is 2-dimensional because it is a list of two numbers. Not surprisingly all 2-dimentional vectors live in a plane. A 3-dimensional vector would be a list of three numbers, and they live in a 3-D volume. A 27-dimensional vector would be a list of twenty- seven numbers, and would live in a space only Ilana’s dad could visualize.
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Magnitudes and direction The “magnitude” of a vector is the distance from the endpoint of the vector to the origin – in a word, it’s length. Suppose we want to calculate the magnitude of the vector !
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This note was uploaded on 08/16/2008 for the course NBIO 228 taught by Professor Greg during the Spring '07 term at Stanford.

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Ch4_Linear_Algebra - Chapter 4 Vectors Matrices and Linear...

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