Introduction to Vector Spaces Linear Algebra, Spring 2011

Transcription

1 Introduction to Vector Spaces Linear Algebra, Spring 2011 You probabl have heard the word vector before, perhaps in the contet of Calculus III or phsics. You probabl think of a vector like this: 5 3 or 2 or a quantit with magnitude and direction These are indeed vectors, but our professors and tetbooks probabl didn t tell ou that there were other kinds of vectors. In fact, the notion of a vector is infinitel more fleible than just those arrows or columns of numbers. As a rough guide, we might offer a tentative, sketch definition : Definition 1 (Rough): A vector is anthing that can be scaled and added. (Scaling means multipling something b a real number to make it larger or smaller, like scaling a recipe up to feed a crowd.) With this rough idea in mind, let us look at two mathematical objects that are eeril similar. 1 Echoes Consider the set P of polnomials with real coefficients, which includes elements such as π and Now we can certainl add and subtract polnomials, and notice that if ou add two polnomials in P together, ou get another polnomial in P. ( π) + ( ) = π P. Moreover, if ou take a real number, sa 6, and multipl it b a polnomial in P, ou get another polnomial in P, scaled b a factor of 6: 6( π) = π P. We summarize these facts b saing that P is closed under addition and scalar multiplication.

2 Vector Spaces Linear Algebra, Spring 2011 Page 2 of 7 Definition 2: Let S be a set. We sa that S is closed under (ordinar) addition if whenever, S, then + S too. We sa that S is closed under (ordinar) multiplication if whenever, S, then S too. We sa that S is closed under scalar multiplication if whenever S and r R, then r S too. Now let s turn our attention to the set of comple numbers C = {a+bi : a, b R}, where i = 1; so C contains elements such as 4, 2i, and 5 6i. Here too, if we add two comple numbers we get another comple number in C. 2i + (5 6i) = 5 4i C And if we take a real number, sa π, and multipl it b an comple number, ou get another comple number: π(5 6i) = 5π (6π)i C So C is also closed under addition and scalar multiplication. Now polnomials are functions, whereas comple numbers are numbers. Those are ver dissimilar widgets, but we found that the have two things in common: in each case ou can add two widgets to get a third widget, and ou can multipl a widget b a real number and get a third widget. If ou re not et impressed b this coincidence, perhaps ou will be after I point out a few more similarities. Either in the case of polnomial-widgets or comple-number-widgets, suppose A, B, and C are widgets. 1. In each case, A + B = B + A. For eample, (5 2 9) + (3 + π) = (3 + π) + (5 2 9) and 2i + (5 6i) = (5 6i) + 2i. 2. In each case, (A + B) + C = A + (B + C). For eample, [ (5 2 9) + (3 + π) ] = (5 2 9) + [ (3 + π) + 2 9] and [4 + (5 6i)] + 2i = 4 + [(5 6i) + 2i]. 3. In each case, a widget doesn t change if ou add zero to it (either the zero polnomial or the comple number 0 = 0 + 0i). For eample, (5 2 9) + 0 = and (5 6i) + (0 + 0i) = 5 6i. 4. In each case, ever widget has an opposite widget, so that when ou add them together ou get zero. For eample, has as its opposite, so that (5 2 9) + (9 5 2 ) = 0, and 5 6i has 6i 5 as its opposite, so that (5 6i) + (6i 5) = 0.

3 Vector Spaces Linear Algebra, Spring 2011 Page 3 of 7 5. In each case, if ou take a real number r, then r(a + B) = ra + rb. For eample, 7 ( (5 2 9)+(3+π) ) = 7(5 2 9)+7(3+π) and 7 ( 2i+(5 6i) ) = 7 2i+7(5 6i). 6. In each case, if ou take two real numbers r and s, then (r + s)a = ra + sa. For eample, (7+2)(5 2 9) = 7(5 2 9)+2(5 2 9) and (7+2)(5 6i) = 7(5 6i)+2(5 6i). 7. In each case, if ou take two real numbers r and s, then r(sa) = (rs)a. For eample, 7 ( 2(5 2 9) ) = (7 2)(5 2 9) and 7 ( 2(5 6i) ) = (7 2)(5 6i). 8. In each case, a widget doesn t change if ou multipl it b the real number 1. For eample, 1(5 2 9) = and 1(5 6i) = 5 6i. The fact that two worlds as different as polnomial functions and comple numbers behave so similarl suggests that we should stud all mathematical sstems in which these ten rules hold. 2 Defining Vector Spaces If we multiplied the polnomial b 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 usuall call the real numbers scalars. The set of scalars, then, is just R, the set of real numbers. 1 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 an single element v V and multipl it b an 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. 1 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 basicall a set in which ou can add, subtract, multipl, and divide. Other fields include the comple numbers C, the rational numbers Q, and even some finite sets. Linear algebra over comple numbers is especiall important to phsicists and electrical engineers. For now, though, ou can just think of scalars as being real numbers.

4 Vector Spaces Linear Algebra, Spring 2011 Page 4 of 7 Vector Space Aioms 0. For all v, w V, v + w V (closure under addition) 00. For all v V and for each scalar r, rv V. (closure under scalar multiplication) 1. For all v, w V, v + w = w + v (commutativit) 2. For all v, w, V, (v + w) + = v + (w + ) (associativit) 3. There eists an element 0 V such that v + 0 = v for all v V. (additive identit) 4. For each v V, there is an element V such that v + = 0. (additive inverses) (We usuall write this as v.) 5. For all v, w V and for each scalar r, r(v + w) = rv + rw (distributive propert) 6. For each v V and for all scalars r and s, (r + s)v = rv + sv (distributive propert) 7. For each v V and for all scalars r and s, r(sv) = (rs)v (multiplicative associativit) 8. For each v V, 1v = v (scalar identit law) Let s be etra 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 ou give me an animal, I can check whether it s a mammal sa, b checking whether it has hair, nurses its oung, etc. Likewise if ou give me a set X an 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 aioms. If the all hold, then X is a vector space; if an aiom fails to hold, then X is not a vector space. If ou tell me something that s true about all mammals sa, that the are warmblooded then I know that statement is true about an particular mammal, such as a lemur. Likewise an theorem we prove from those ten aioms will be true about all vector spaces, so it will be true about an particular vector space, such as C or P. This is the great power of abstraction that makes mathematics so amazing: without seeing individual vector spaces, working merel from the ten aioms the have in common, we can prove beautiful and powerful facts that are true for all of them. A note on notation One of the great dangers in linear algebra is getting confused between scalars and vectors. After all, scalars are real numbers from R, whereas vectors are something else entirel mabe polnomials or comple numbers or arrows or points or heaven-knows-what. To help keep ourselves from getting confused, there are two notational strategies. First, we ll alwas print vectors in boldface (like u and v) and scalars in regular italic tpe (like r and s). In handwritten work (like on the whiteboard) we will write a little arrow over the letter, like this: u and v. Second, we will tr to choose our vectors names from the end of the alphabet, especiall u, v, w,,, and z. Scalars will come from earlier in the alphabet, like r, s, and t or a, b, and c. It s especiall important to remember the distinction between 0, which is the real number zero we grew up with, and 0, which is a vector in our vector space V. Remember, 0 0!! We will usuall use capital letters to denote sets, matrices, or linear transformations (whatever those are).

5 Vector Spaces Linear Algebra, Spring 2011 Page 5 of 7 3 Euclidean Space (R n ) As mentioned before, if ou worked with vectors before, it was probabl with vectors like 5 3 or 2 or a quantit with magnitude and direction In those das ou were working with one of the most important families of vector spaces there is. Let s start with a simple eample. 3.1 R 2 Consider the set of ordered pairs of real numbers (, ), and define addition and scalar multiplication b ( 1, 1 ) + ( 2, 2 ) = ( 1 + 2, ) and r( 1, 1 ) = (r 1, r 1 ). We could think of this (, ) as an arrow on the -plane stretching from the origin (where the - and -aes cross) to a point units to the right and units up. For eample, the following figure shows the vectors (2, 3) and (4, 1): (2, 3) (4, 1) In this interpretation, scaling a vector corresponds to adjusting its length: v = (4, 1) 1 v 2 2v Addition of can be pictured in two was. First, ou can la them end to end, the tip of one vector at the tail of the other, and see where the point. Second, ou can leave them rooted at the origin, but construct the parallelogram and find the far corner. The net figure shows that (2, 3) + (4, 1) = (6, 4) using these two methods.

6 Vector Spaces Linear Algebra, Spring 2011 Page 6 of 7 The vector (4, 1) shifted up. (6, 4) (2, 3) Parallelogram method. (6, 4) (2, 3) (4, 1) (4, 1) The set of ordered pairs of real numbers is written as R 2. It is ver eas to check that the set R 2, with addition and scalar multiplication defined in this wa, does satisf all ten aioms, so it is a vector space. You are asked to do so in Eercise 3d. 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 (,, z), such as (3, 4, 2) and (5, 3, 2). The rules for addition and scalar multiplication are essentiall the same: ( 1, 1, z 1 ) + ( 2, 2, z 2 ) = ( 1 + 2, 1 + 2, z 1 + z 2 ) and r( 1, 1, z 1 ) = (r 1, r 1, rz 1 ). The set of ordered triples of real numbers is, naturall, called R 3. You should be able to verif that R 3, with this definition of addition and scalar multiplication, is also a vector space. Verif Aioms (4), (5), and (7) now for practice. 3.3 R n B now ou probabl have realized that there is nothing special about having two (in R 2 ) or three (in R 3 ) real numbers in each vector. Each of our proofs that the aioms hold would work just as well if instead of ordered pairs or triples, we had ordered quintuples or septuples or an multiple at all. Let n be a positive integer. Then b 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 perfectl analogous wa: (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 ). Unfortunatel, we can t reall 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 horizontall like (,, z), we could write them verticall like if we preferred. As long as we don t get confused, z especiall later on when we start multipling b matrices, there s no harm in writing a vector either wa.

7 Vector Spaces Linear Algebra, Spring 2011 Page 7 of 7 Eercises 1. Determine whether each of the following sets is closed under ordinar addition. If so, justif our answer. If not, give a specific countereample. (Recall interval notation: b [2, 4] we mean { R : 2 4}, etc.) (a) [5, ) (b) [2, 4] (c) {3, 6, 9, 12,...} (d) {0} 2. Determine whether each of the following sets is closed under ordinar multiplication. If so, justif our answer. If not, give a specific countereample. (a) (, 0] (b) { 1, 0, 1} (c) [0, 1] (d) [0, 2] (e) {3, 6, 9, 12,...} (f) [ 1, 1] 3. Determine whether each of the following is a vector space b checking the 10 aioms. (Unless specified otherwise, addition and scalar multiplication are defined in the natural wa.) If it is, justif wh each of the ten aioms holds. If not, give at least one aiom that fails. (a) The set of all polnomials with integer coefficients. (b) The set of all real numbers. (c) The subset [0, ) of the number line. (d) The set of ordered pairs {(, ) :, R} with addition defined b and scalar multiplication defined b ( 1, 1 ) + ( 2, 2 ) = ( 1 + 2, ) r(, ) = (r, r). 4. (a) What is the smallest subset of R that contains 1 2 (b) What is the smallest subset of R that contains 1 2 multiplication? and is closed under addition? and is closed under ordinar 5. Let V be a vector space, let r R, and let v V. Eplain how the minus sign is used differentl in each of the following three epressions: ( r)v, r( v), and (rv). 6. Using our initial, rough definition of a vectors as things ou can scale and add, come up with another vector space besides the ones ou have seen. (Tr to be creative.)