Introduction to Vector Spaces Linear Algebra, Fall 2008 1 Echoes Consider the set P of polynomials with real coefficients, which includes elements such as 7x 3 4 3 x + π and 3x4 2x 3. Now we can add, subtract, multiply, and divide polynomials, but notice that if you add two polynomials in P together, you get another polynomial in P. (7x 3 4 3 x + π) + (3x4 2x 3 ) = 3x 4 + 5x 3 4 3 x + π P. Moreover, if you take a real number, say 6, and multiply it by a polynomial in P, you get another polynomial in P. 6(7x 3 4 3 x + π) = 42x3 8x + 6π P. We can also think about the set of complex numbers C = {a + bi : a, b R}, which contains elements such as 4, 2i, and 5 6i. Here too, if we add two complex numbers we get another complex number in C. 2i + (5 6i) = 5 4i C And if we take a real number, say π, and multiply it by any complex number, you get another complex number: π(5 6i) = 5π (6π)i C Now polynomials are functions, whereas complex numbers are numbers. Those are very dissimilar widgets, but we found that they have two things in common: in each case you can add two widgets to get a third widget, and you can multiply a widget by a real number and get a third widget. If you re not yet impressed by this coincidence, perhaps you will be after I point out a few more similarities. Either in the case of polynomial-widgets or complex-number-widgets, suppose A, B, and C are widgets. 3. In each case, A + B = B + A. For example, (5x 2 9) + (3x + π) = (3x + π) + (5x 2 9) and 2i + (5 6i) = (5 6i) + 2i. 4. In each case, (A + B) + C = A + (B + C). For example, [ (5x 2 9) + (3x + π) ] + 2x 9 = (5x 2 9) + [ (3x + π) + 2x 9] and [4 + (5 6i)] + 2i = 4 + [(5 6i) + 2i]. 5. In each case, a widget doesn t change if you add zero to it (either the zero polynomial or the complex number 0 = 0 + 0i). For example, (5x 2 9) + 0 = 5x 2 9 and (5 6i) + (0 + 0i) = 5 6i.
Vector Spaces Linear Algebra, Fall 2008 Page 2 of 6 6. In each case, every widget has an opposite widget, so that when you add them together you get zero. For example, 5x 2 9 has 9 5x 2 as its opposite, so that (5x 2 9) + (9 5x 2 ) = 0, and 5 6i has 6i 5 as its opposite, so that (5 6i) + (6i 5) = 0. 7. In each case, if you take a real number r, then r(a + B) = ra + rb. For example, 7((5x 2 9)+(3x+π)) = 7(5x 2 9)+7(3x+π) and 7(2i+(5 6i)) = 7 2i+7(5 6i). 8. In each case, if you take two real numbers r and s, then (r + s)a = ra + sa. For example, (7+2)(5x 2 9) = 7(5x 2 9)+2(5x 2 9) and (7+2)(5 6i) = 7(5 6i)+2(5 6i). 9. In each case, if you take two real numbers r and s, then r(sa) = (rs)a. For example, 7 [ 2(5x 2 9) ] = (7 2)(5x 2 9) and 7 [2(5 6i)] = (7 2)(5 6i). 10. In each case, a widget doesn t change if you multiply it by the real number 1. For example, 1(5x 2 9) = 5x 2 9 and 1(5 6i) = 5 6i. The fact that two such different worlds as polynomial functions and complex numbers behave so similarly suggests that we should study all mathematical systems in which these ten rules hold. 2 Defining Vector Spaces Scalars. If we multiplied the polynomial 5x 2 9 by 2, we would double it, as though we were scaling a recipe up to feed a larger crowd. For this reason, when we re doing linear algebra we usually call the real numbers scalars. The set of scalars, then, is just R. (In fact, it s possible to do linear algebra with other sets of scalars, as long as the set of scalars is what mathematicians call a field basically a set in which you can add, subtract, multiply, and divide. Other fields include the complex numbers C, the rational numbers Q, and even some finite sets. For now, though, you can just think of scalars as being real numbers.) Let V be a set, and suppose we have two rules or operations for transforming elements of V. First, we can take two elements v and w of V and put them together using an operation we ll call + to get a new object called v + w. Second, we can take any single element v V and multiply it by any scalar r to get a new object rv. If the following ten rules hold, then we will call V a vector space, and we will call each element of V a vector.
Vector Spaces Linear Algebra, Fall 2008 Page 3 of 6 Vector Space Axioms 1. For all u, v V, u + v V (closure under addition) 2. For all u, v V, u + v = v + u (commutativity) 3. For all u, v, w V, (u + v) + w = u + (v + w) (associativity) 4. There exists an element 0 V such that u + 0 = u for all u V. (additive identity) 5. For each u V, there is an element u V (additive inverses) such that u + ( u) = 0. 6. For all u V and for any scalar c, cu V. (closure under scalar multiplication) 7. For all u, v V and for any scalar c, c(u + v) = cu + cv (distributive property) 8. For any u V and for all scalars c and d, (c + d)u = cu + du (distributive property) 9. For any u V and for all scalars c and d, c(du) = (cd)u (multiplicative associativity) 10. For any v V, 1u = u (scalar identity law) Let s be extra clear about what this means. Some mathematical objects are vector spaces and some are not, just like some animals are mammals and some are not. If you give me an animal, I can check whether it s a mammal say, by checking whether it has hair, nurses its young, etc. Likewise if you give me a set X any old set of mathematical objects at all and tell me how to do addition and scalar multiplication on that set, then I can check those ten axioms. If they all hold, then X is a vector space; if any axiom fails to hold, then X is not a vector space. If you tell me something that s true about all mammals say, that it is warm-blooded then I know that statement is true about any particular mammal, say a lemur. Likewise any theorem we prove from those ten axioms will be true about all vector spaces, so it will be true about any particular vector space, such as C or the set of polynomials. This is the great power of abstraction that makes mathematics so amazing: without seeing every single vector space, from the ten axioms they have in common we can prove beautiful and powerful facts that are true for all of them. 3 Euclidean Space (R n ) If you have heard the word vector before, perhaps in the context of an advanced calculus class, you may have been confused during the last section, since you perhaps expected a vector to look like this: 5 v or 3 or 5 i + 3 j 2 k or 2 or a quantity with magnitude and direction In those days you were working with one of the most important families of vector spaces there is, but your professors and textbooks probably didn t tell you that there were other kinds of vector spaces.
Vector Spaces Linear Algebra, Fall 2008 Page 4 of 6 Let s start with a simple example. 3.1 R 2 Consider the set of ordered pairs of real numbers (x, y), and define addition and scalar multiplication by (x 1, y 1 ) + (x 2, y 2 ) = (x 1 + x 2, y 1 + y 2 ) and r(x 1, y 1 ) = (rx 1, ry 1 ). We could think of this (x, y) as an arrow on the xy-plane stretching from the origin (where the x- and y-axes cross) to a point x units to the right and y units up. For example, the following figure shows the vectors (2, 3) and (4, 1): In this interpretation, we add two 5 x:\file.dat using 1:2:3:4 4 3 (2,3) 2 1 (4,1) 0-1 -1 0 1 2 3 4 5 6 7 vectors by laying them head to tail and seeing where they point. The next figure shows that (2, 3) + (4, 1) = (6, 4). The set of ordered pairs of real numbers is written as R 2. It is very 5 x:\r2v2.dat using 1:2:3:4 4 (6,4) 3 (2,3) 2 1 0-1 -1 0 1 2 3 4 5 6 7 easy to check that the set R 2, with addition and scalar multiplication defined in this way, does satisfy all ten axioms, so it is a vector space.
Vector Spaces Linear Algebra, Fall 2008 Page 5 of 6 3.2 R 3 We can likewise think of arrows stretching from the origin in three-dimensional space, which can be written as ordered triples of real numbers (x, y, z), such as (3, 4, 2) and (5, 3, 2). The rules for addition and scalar multiplication are essentially the same: (x 1, y 1, z 1 ) + (x 2, y 2, z 2 ) = (x 1 + x 2, y 1 + y 2, z 1 + z 2 ) and r(x 1, y 1, z 1 ) = (rx 1, ry 1, rz 1 ). The set of ordered triples of real numbers is, naturally, called R 3. You should be able to verify that R 3, with this definition of addition and scalar multiplication, is also a vector space. Do so now for practice. 3.3 R n If you have verified the ten axioms for R 2 and R 3, you probably have realized that there was nothing special about there being two or three real numbers in each vector. Each of your proofs that the axioms hold would work just as well if instead of ordered pairs or triples, we had ordered quintuples or septuples or any multiple at all. Let n be a positive integer. Then by R n we denote the set of ordered n-tuples R n = {(v 1, v 2,..., v n ) : v 1,..., v n R}. We define addition and scalar multiplication in a perfectly analogous way: (v 1,..., v n ) + (w 1,..., w n ) = (v 1 + w 1,..., v n + w n ) and r(v 1,..., v n ) = (rv 1,..., rv n ). Unfortunately, we can t really visualize R n if n > 3, but that shouldn t stop us from doing mathematics in it. One final note: instead of writing our vectors horizontally like x (x, y, z), we could write our vectors vertically like y if we preferred. As long as we don t z get confused, especially later on when we start multiplying by matrices, there s no harm in writing a vector either way. Exercises 1. Determine whether each of the following sets is closed under ordinary addition. If so, justify your answer. If not, give a counterexample. (Recall interval notation: by [2, 4] we mean {x R : 2 x 4}, etc.) (a) [5, ) (b) [2, 4] (c) {3, 6, 9, 12,...} (d) {0} (e)
Vector Spaces Linear Algebra, Fall 2008 Page 6 of 6 2. Determine whether each of the following sets is closed under ordinary multiplication. If so, justify your answer. If not, give a counterexample. (a) (, 0] (b) [0, ) (c) Z, the set of all integers (d) {3, 6, 9, 12,...} (e) { 1, 0, 1} (f) [ 1, 0] (g) [0, 1] (h) [1, 2] (i) [ 1, 1] (j) [0, 2] 3. Determine whether each of the following is a vector space. (Unless specified otherwise, define addition and scalar multiplication are defined in the natural way.) If not, list all of the axioms it fails to satisfy. (a) The set of all polynomials with integer coefficients. (b) The set of all continuous functions. (c) The set of all real numbers. (d) The subset [0, ) of the number line. (e) The set R 2 of ordered pairs (a, b) of real numbers, with scalar multiplication defined as usual but with a funny addition defined by (a, b) + (c, d) = (a + c, 0) (f) The set R 3 of ordered triples (x, y, z), with addition defined as usual but with a bizarre scalar multiplication defined by r(x, y, z) = (rx, ry, z) (g) The set (h) The set (i) The set {(x, y, 1) R 3 } {(x, y, 0) R 3 } {(x, y, z) R 3 : x + y + z = 0}