Information about Abstract Linear Algebra - Part 1

00:00 Introduction

00:28 Prerequisites

01:28 Overview

03:40 Definition of a vector space

05:56 Operations on vector spaces

08:10 8 Rules on these operations

1 00:00:00,386 –> 00:00:04,224 Hello and welcome to abstract linear algebra.

2 00:00:04,314 –> 00:00:07,972 The video course that extends my other linear algebra course.

3 00:00:08,514 –> 00:00:16,307 and before we start, as always I want to give a big thank you to all supporters on Steady, here on YouTube or on Patreon.

4 00:00:16,507 –> 00:00:21,280 You make it possible that I can create such videos about mathematics

5 00:00:21,480 –> 00:00:27,660 and there you should know as a supporter, you can download PDF versions and quizzes for all the videos.

6 00:00:28,186 –> 00:00:34,436 Ok, then let’s immediately start with this new series by telling you about the prerequisites you need.

7 00:00:34,636 –> 00:00:41,144 As always my “start learning mathematics” course gives you a good foundation for all my other courses.

8 00:00:41,571 –> 00:00:47,736 This is what you need, when you want to know how to work with sets and how to construct the numbers sets

9 00:00:48,229 –> 00:00:54,901 and based on that knowledge you can immediately go to my “linear algebra” course or to my “real analysis” course

10 00:00:55,101 –> 00:01:03,185 and know it might not surprise you that watching my “linear algebra” course really helps understanding this “abstract linear algebra” course.

11 00:01:03,657 –> 00:01:11,467 However you don’t need the full knowledge of this concrete linear algebra course to understand our new abstract linear algebra course,

12 00:01:11,857 –> 00:01:19,809 but you need the core concepts from calculating in R^n and C^n to get the idea how to generalize these.

13 00:01:20,414 –> 00:01:28,671 Moreover it also helps to have some knowledge of real analysis such that you can understand some abstract examples here.

14 00:01:29,043 –> 00:01:36,067 Ok, after this I think I can give you a short overview of what to expect in this video series here.

15 00:01:36,457 –> 00:01:42,840 In some sense you can say, we will generalize everything we already know from R^n and C^n.

16 00:01:43,040 –> 00:01:48,342 This means from now on we will work in general or abstract vector spaces

17 00:01:48,542 –> 00:01:54,552 and then R^n or C^n are just some special examples for vector spaces.

18 00:01:54,752 –> 00:01:59,510 Moreover in the same sense we can also generalize matrices.

19 00:01:59,710 –> 00:02:07,865 So you should already know, matrices are related to linear maps and here we will define abstract, general linear maps.

20 00:02:08,571 –> 00:02:16,104 Therefore one important topic here is if we can still use matrices to calculate with such linear maps

21 00:02:16,304 –> 00:02:21,112 and related to this is an important concept we call change of basis.

22 00:02:21,312 –> 00:02:29,734 Indeed, we can define the notion basis for an abstract vector space and then the question is: “what happens when we change it?”

23 00:02:29,934 –> 00:02:37,624 In fact we already discussed it a little bit in the old course, but now we can discuss it in all generality.

24 00:02:37,824 –> 00:02:43,379 Also on an abstract level we will talk about angles and lengths

25 00:02:43,579 –> 00:02:49,720 and there you might already know, in order to do that we need the concept of a general inner product.

26 00:02:50,029 –> 00:02:57,458 There we will see that with an inner product we have much more geometry in such an abstract vector space.

27 00:02:57,658 –> 00:03:04,294 In fact we need the inner product such that we can say that 2 objects are orthogonal to each other

28 00:03:04,494 –> 00:03:09,106 and we will see that we have a lot of important applications for that.

29 00:03:09,306 –> 00:03:14,637 Ok and now the last topic I want to drop here is the one about eigenvalues.

30 00:03:14,957 –> 00:03:22,717 There you might already know eigenvalues for matrices and now we can translate this to general linear maps.

31 00:03:22,917 –> 00:03:31,335 Ok, so in summary we can say, what we will do is that we generalize all the notions we already know from the concrete level.

32 00:03:31,414 –> 00:03:39,737 Moreover I would say after doing that, you will see why linear algebra is such a powerful tool in all parts of mathematics

33 00:03:40,329 –> 00:03:45,248 and then I would say let’s start with the first definition of this course

34 00:03:45,743 –> 00:03:50,067 and this is not so surprising. This will be the definition of a vector space

35 00:03:50,543 –> 00:03:58,558 and in order to understand this definition, just imagine what we already know from the vector spaces R^n and C^n.

36 00:03:58,758 –> 00:04:03,023 In fact these are vector spaces, because of 2 operations.

37 00:04:03,071 –> 00:04:06,317 First we know that we can add 2 vectors

38 00:04:06,517 –> 00:04:11,939 and as always we can visualize this operation by putting the arrows together

39 00:04:12,139 –> 00:04:17,166 and moreover the second operation is that we can scale a vector

40 00:04:17,366 –> 00:04:22,279 and with that we have exactly what we want. We want to calculate with vectors.

41 00:04:22,479 –> 00:04:25,959 Namely we want to add them and we want to scale them.

42 00:04:26,159 –> 00:04:30,033 So we need vector addition and scalar multiplication

43 00:04:30,233 –> 00:04:36,431 and here please don’t forget for the second operation here, we need scalars as scaling factors

44 00:04:36,843 –> 00:04:41,406 and in fact they always have to come from a so called field

45 00:04:41,606 –> 00:04:49,723 and of course for R^n or C^n this is not so complicated, because they come either from the real number line or from the complex plane

46 00:04:49,923 –> 00:04:57,438 and there you should know both are fields, so you can calculate in them with the normal calculation rules for numbers.

47 00:04:57,638 –> 00:05:01,757 In particular you know how to add numbers. You can multiply them.

48 00:05:01,814 –> 00:05:07,785 We have a neutral element with respect to the addition and another one with respect to the multiplication and so on

49 00:05:08,314 –> 00:05:13,177 and more details about that, you find in my “start learning numbers” series.

50 00:05:13,543 –> 00:05:20,540 The important fact here is just that we need ordinary numbers, for the scaling factors in our vector space.

51 00:05:21,043 –> 00:05:27,415 Ok, but now we just want to put these 2 operations in the definition of a vector space.

52 00:05:27,615 –> 00:05:36,484 In other words we will take a set V such that we can calculate with the elements in it as we calculate in R^n or C^n.

53 00:05:36,586 –> 00:05:41,778 Therefore the first thing we want to fix here is a field F.

54 00:05:41,978 –> 00:05:48,952 This now could be any field, but often it’s either the real number line or the complex plane again.

55 00:05:49,152 –> 00:05:53,665 Of course the choice of the field definitely depends on which problem you want to tackle,

56 00:05:53,865 –> 00:05:57,485 but for the moment it’s enough to think of R or C.

57 00:05:57,914 –> 00:06:03,207 However now the vector space V can be any set that is not the empty set.

58 00:06:03,900 –> 00:06:08,666 This is what you should immediately remember. A vector space can never be empty.

59 00:06:08,866 –> 00:06:15,463 Ok and now by what we have said before, we know that we need exactly 2 operations for this set.

60 00:06:15,600 –> 00:06:19,757 This simply means that we have 2 maps in the definition.

61 00:06:20,098 –> 00:06:25,134 The first one is the vector addition we usually denote with a plus sign.

62 00:06:25,334 –> 00:06:29,093 However, please don’t forget, this is now an abstract plus sign.

63 00:06:29,529 –> 00:06:34,269 So definitely operation than the addition in R or C.

64 00:06:34,469 –> 00:06:40,451 So now we know, the vector addition needs 2 inputs. Namely 2 vectors from V

65 00:06:40,971 –> 00:06:44,774 and then the output should be again a vector in V.

66 00:06:45,186 –> 00:06:49,610 So not so fancy. 2 vectors together gives us a new vector.

67 00:06:49,957 –> 00:06:54,568 Moreover on the other hand we also have the scalar multiplication

68 00:06:55,086 –> 00:06:59,107 and maybe not so surprising this one we will denote with a dot.

69 00:06:59,857 –> 00:07:03,515 However again here it’s an abstract multiplication sign.

70 00:07:04,057 –> 00:07:10,362 So here again we have 2 inputs, but the first one is now the field.

71 00:07:10,562 –> 00:07:15,336 Namely we put in one scalar and one vector from V

72 00:07:15,536 –> 00:07:20,211 and not so surprising, what should come out is a vector in V again.

73 00:07:20,529 –> 00:07:27,024 Now you should see, we immediately know examples for these operations when you look at R^n or C^n.

74 00:07:27,224 –> 00:07:33,640 However from these examples we know that the 2 operations also fulfill some rules.

75 00:07:33,840 –> 00:07:39,225 Indeed, if we put them together, we see we need exactly 8 rules.

76 00:07:39,425 –> 00:07:45,336 In other words, if these rules are satisfied, we can call V a vector space.

77 00:07:45,536 –> 00:07:49,494 More precisely we will call it a F vector space.

78 00:07:50,286 –> 00:07:56,147 So here please don’t forget, the corresponding field for the scalars is included in the definition.

79 00:07:56,347 –> 00:08:03,080 So for example if the corresponding field is given by the real numbers, you would say we have a real vector space

80 00:08:03,280 –> 00:08:10,311 and as I said before at least in this course real and complex vector spaces are the most important ones.

81 00:08:10,971 –> 00:08:16,735 Ok, but now as you might expect it is time to list the 8 rules we need.

82 00:08:17,129 –> 00:08:26,553 Indeed, the first 4 we can already put together, by just saying that V together with the vector addition is a so called abelian group.

83 00:08:26,753 –> 00:08:33,644 This means we have the 3 group axioms and in addition the operation is also commutative.

84 00:08:34,071 –> 00:08:38,741 More precisely the first property of the group is associativity.

85 00:08:38,941 –> 00:08:46,080 This means if we take elements u, v, w from V, we can exchange the parentheses like that.

86 00:08:46,280 –> 00:08:53,865 Ok, moreover you might also know the second rule is that we have a neutral element with respect to the addition.

87 00:08:54,065 –> 00:08:58,977 Of course this is then what we call the 0 vector in our abstract vector space

88 00:08:59,177 –> 00:09:04,891 and then with respect to this neutral element we also find inverse elements.

89 00:09:05,571 –> 00:09:14,109 More precisely it means that for each lower case v in our vector space V, we find another vector we call -v

90 00:09:14,171 –> 00:09:17,425 such that the addition gives us the 0 vector.

91 00:09:17,625 –> 00:09:24,242 So the requirement here is that all the inverses with respect to addition also lie in our set V.

92 00:09:24,943 –> 00:09:30,718 Ok and then the last rule I already told you about. It simply means that the addition is commutative.

93 00:09:30,918 –> 00:09:35,717 Hence it does not matter in which order we add up 2 vectors.

94 00:09:36,229 –> 00:09:41,996 Ok, then let’s go to the next 2 rules, which concerns the scalar multiplication

95 00:09:42,529 –> 00:09:48,028 and we can summarize them by saying that the scalar multiplication is compatible

96 00:09:48,386 –> 00:09:52,600 and first that means the scalar multiplication acts nicely

97 00:09:52,714 –> 00:09:57,200 when we put it together with the multiplication we already know from our field.

98 00:09:57,400 –> 00:10:05,113 So you should see, here on the left-hand side we scale the vector v 2 times, but on the right-hand side we only scale it once.

99 00:10:05,714 –> 00:10:09,643 However the claim here is that the result should be the same vector.

100 00:10:09,740 –> 00:10:14,036 Which means you can multiply the scalars beforehand.

101 00:10:14,236 –> 00:10:19,491 So it means the scalar multiplication is compatible with the other multiplication

102 00:10:19,691 –> 00:10:22,886 and moreover the second rule here tells us

103 00:10:22,971 –> 00:10:29,123 that the multiplicative unit from our field is also compatible with this scalar multiplication.

104 00:10:29,323 –> 00:10:34,667 So if you multiply with the scalar 1, you don’t change the vector v at all.

105 00:10:34,867 –> 00:10:38,749 Ok and now we exactly need 2 rules more.

106 00:10:38,949 –> 00:10:44,149 Also these can be put together as we simply speak of distributive laws

107 00:10:44,349 –> 00:10:51,113 and of course they are essential, because they connect the scalar multiplication with the vector addition.

108 00:10:51,313 –> 00:10:59,132 So for example, here it means scaling the result of a vector addition is the same as adding the two scaled vectors

109 00:10:59,714 –> 00:11:07,600 and on the other hand if we first add 2 scalars in our field F, it’s the same as adding the scaled vectors.

110 00:11:08,114 –> 00:11:14,368 So you see, here it’s important to note that the two plus signs denote two different operations.

111 00:11:14,814 –> 00:11:22,740 However, now we have learned that the rules of the vector space claim that the two different operations act nicely together.

112 00:11:22,940 –> 00:11:25,854 Ok, very nice. Now we have everything.

113 00:11:26,054 –> 00:11:29,618 The result that we get here is an abstract vector space

114 00:11:30,186 –> 00:11:38,456 and with these rules now, we can calculate inside this abstract vector space, as we already know it from R^n and C^n.

115 00:11:38,857 –> 00:11:45,678 This means the visualization we have with these arrows also works in this abstract sense.

116 00:11:46,129 –> 00:11:52,974 This means imagining vectors given as arrows always helps to remember these operations.

117 00:11:53,174 –> 00:11:58,586 However, you should always keep in mind that the elements of a vector space are not arrows.

118 00:11:58,614 –> 00:12:01,057 We only use them for the visualization.

119 00:12:01,571 –> 00:12:08,020 Indeed, the elements of a vector space, so called vectors, could be now very abstract objects

120 00:12:08,629 –> 00:12:13,919 and I can already promise you, in the next video we will see such nice examples.

121 00:12:14,119 –> 00:12:19,229 Therefore I really hope we meet again and have a nice day. Bye bye!

Q1: Is $\mathbb{R}$ with the ordinary addition and multiplication a real vector space?

A1: Yes, it is.

A2: No, the multiplication is not a scalar multiplication.

A3: No, the addition is missing a neutral element.

A4: One needs more information.

Q2: Consider a set with one element ${ a }$ together with the addition $a + a = a$ and the scalar multiplication $\lambda \cdot a = a$ for all $\lambda \in \mathbb{C}$. Does this give a complex vector space?

A1: Yes, it does.

A2: No, the inverses with respect to the addition are missing.

A3: No, the distributive laws are not satisfied.

A4: One needs more information.

Q3: Consider a complex vector space $V$. Let’s count the elements of $V$. What is not possible?

A1: $V$ has exactly $2$ elements.

A2: $V$ has exactly $1$ element.

A3: $V$ has infintely many elements.

A4: $V$ has finitely many elements.