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1 Welcome to the World of Linear Algebra: Vector Spaces Engineering Mathematics 1{Summer 2012 Linear Algebra The commercial message. Linearity is a key concept in mathematics and its applications.Linear objects are usually nice, smooth, clean, easy to handle. Nonlinear ones are slimy, temperamental, and full of dark corners in which monsters can hide. The world, unfortunately, has a strong tendency toward non-linearity, but there is no way that we can understand nonlinear phenomena without first having a good understanding of linear ones. In fact, one of the most common first approaches to a non-linear problem may be to approximate it by a linear one. 1 Welcome to the world of linear algebra: Vector Spaces Vector spaces, also known as a linear spaces, come in two flavors, real and complex. The main difference between them is what is meant by a scalar. When working with real vector spaces, a scalar is a real number. When working with complex vector spaces, a scalar is a complex number. The important thing is not to mix the two flavors. You either work exclusively with real vector spaces, or exclusively with complex ones. Well . ; nothing is that definite. There are times when one wants to go from one flavor to the other; but that should be done with care. One thing to remember, however, is that real numbers are also part of the complex universe; a real number is just a complex number with zero imaginary part. So when working with complex vector spaces, real numbers are also scalars because they are also complex numbers. So what is a vector space? We could give a vague definition saying it is an environment that is linear algebra friendly. But we need to be more precise. Linear operations are very basic ones, so as a better definition we can say that a vector space is any set of objects in which we have defined two operations: 1. An addition, so that if we have any two objects in that set, we know what their sum is. And that sum should be also an element of the set. 2.A scalar multiplication; it should be possible to multiply objects of the set by scalars to get (usually) new objects in the set. If by scalars we mean real numbers, then we have a real vector space; if we mean complex numbers, we have a complex vector space. There is some freedom in how these operations are defined, but only a little bit. The operations should have the usual properties we associate with sums and products. Before giving examples, we need to be a bit more precise. In general, I will use lower case boldface letters to stand for vectors, a vector being simply an object in a set that has been identified as a vector space, and regularly written lower case letters for scalars (real or complex numbers). If V is a vector space, if v; w are elements of V (in other words, vectors), then their sum is denoted by (written as) v + w, which also has to be in V . If a is a scalar, v an element of V , then av denotes the scalar product of a times v. It should be an element of V . Summary of the information so far. By scalar we mean a complex or a real number. We should be aware that if our scalars are real numbers, they should always be real; if complex, always complex. A vector (or linear) space is any set of objects in which we have defined a sum and a scalar product satisfying some (soon to be specified) properties. In this context, the elements of this set are called vectors. So we can say that a vector set is a bunch of vectors closed under addition and scalar multiplication, a vector is an element of a vector space. Finally!, here are the basic properties these operations have to satisfy. For a set V of objects to deserve to be called a vector space it is, in the first place necessary (as mentioned already several times) that if v, w 2 V , then we have defined v + w 2 V , and if a is a scalar, v 2 V , we have defined av 2 V . These operations must satisfy: 1. (Associativity of the sum) If v; w; z 2 V , then (v + w) + z = v + (w + z). This allows us to write the sum of any number of objects (vectors) without using parentheses: If v; w; y; z 2 V , it makes sense to write v + w + y + z. One person may compute this sum by adding first v + w, then y + z, finally adding these 1 WELCOME TO THE WORLD OF LINEAR ALGEBRA: VECTOR SPACES 2 two together. Another person may first compute w + y, then add v in front to get v + (w + y), finally add z to get (v + (w + y)) + z. The result is the same. 2. (Commutativity of the sum) If v; w 2 V , then v + w = w + v. 3. (Existence of 0) There exists a unique element of V , usually denoted by 0, such that v + 0 = v, for all v 2 V . 4. (Existence of an additive inverse) If v 2 V , there exists a unique element −v 2 V such that v + (−v) = 0. This allows us to define subtraction; one writes v − w for v + (−w). 5. (Distributivity 1) If v; w 2 V , a 2 R, then a(v + w) = av + aw. 6. (Distributivity 2) If a; b 2 R, and v 2 V , then (a + b)v = av + bv. 7. (Associativity 2) If a; b 2 R, v 2 V , then a(bv) = (ab)v = b(av). 8. (One is one) 1v = v for all v 2 V . (Here 1 is the scalar number 1.) These properties have consequences. The main consequence is that one operates with vectors as with numbers, as long as things make sense. So, for example, in general, one can't multiply a vector by a vector (there will be particular exceptions). But otherwise, what seems right is usually right. Here are some examples of what is true in any vector space V . Any scalar times the 0 vector is the zero vector. In symbols: a0 = 0. It is not one of the eight properties listed above, but it follows from them, It is also true that the scalar 0 times any vector is the zero vector: 0v = 0. One good thing is that all the linear spaces we will consider will be quite concrete, and all the properties of the operations quite evident. At least, so I hope. What this abstract part does is to provide a common framework for all these concrete sets of objects. Here are a number of examples. It might be a good idea to check that in every case the 8 properties hold or, at the very least, convince yourself that they hold. • Example 1. The stupid vector space. (But, since we are scientists and engineers, we cannot use the word stupid. We must be dignified. It is usually called the trivial vector space.) It is a silly example, but it needs to be seen. It is the absolute simplest case of a linear space. The space has a single vector, the 0 vector. Addition is quite easy; all you need to know is that 0 + 0 = 0. So is scalar multiplication: If a is a scalar, then a0 = 0. And −0 = 0. The trivial vector space can be either real or complex. The next set of examples consist of real vector spaces. • Example 2. The next vector space, just one degree above the previous one in complexity, is the set R of real numbers. Here the real numbers are forced to play a double role, have something like a double personality: On the one hand they are the numbers they always are, the scalars. But they also are the vectors. If a; b 2 R (I write them in boldface to emphasize that now they are acting as vectors), one defines a + b as usual. And if a 2 R and c is a scalar, then ca is just the usual product ca of c times a, but now interpreted as a vector. • Example 3. Our main examples of real vector spaces are the spaces known as Rn, where n is a positive integer. We already saw R1; it is just R. Now we will meet the next one: the space R2 consists of all pairs of real numbers; in symbols: 2 R = f(a; b): a; b are real numbersg: As we all know, we can identify the elements of R2 with points in the plane; once we fix a system of cartesian coordinates, we identify the pair (a; b) with the point of cartesian coordinates (a; b). Operations are defined in a more or less expected way: (a; b) + (c; d) = (a + c; b + d); (1) a(c; d) = (ac; ad): (2) Verifying associativity, commutativity, and distributivity, reduces to the usual associative, commutative, and distributive properties of the operations for real numbers. The 0 element is 0 = (0; 0); obviously (a; b) + 0 = (a; b) + (0; 0) = (a + 0; b + 0) = (a; b): 1 WELCOME TO THE WORLD OF LINEAR ALGEBRA: VECTOR SPACES 3 The additive inverse is also easy to identify, −(a; b) = (−a; −b): (a; b) + (−a; −b) = (a + (−a); b + (−b)) = (0; 0) = 0: These operations have geometric interpretations. I said that R2 can be identified with points of the plane. But another interpretation is to think of the elements of R2 as being vectors, we represent a pair (a; b) as an arrow beginning at(0; 0) and ending at the point of coordinates (a; b).
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