00:00 Introduction

00:50 Linear Algebra applications

02:20 Visit to the abstract level

03:00 Concrete level

03:40 Prerequisites

04:10 Credits

1 00:00:00,880 –> 00:00:06,560 Hello and welcome to linear algebra A whole video course I designed for everyone 

2 00:00:06,560 –> 00:00:12,160 who wants to start with the basics in mathematics and in the end wants to master vectors,  

3 00:00:12,160 –> 00:00:17,760 vector spaces, matrices and so on. However, before we start I really  

4 00:00:17,760 –> 00:00:21,280 want to thank the nice supporters that make this course possible. 

5 00:00:22,080 –> 00:00:26,480 Each small contribution on Steady or  PayPal makes it easier for me to spend 

6 00:00:26,480 –> 00:00:28,400 to create these videos here. 

7 00:00:29,360 –> 00:00:34,480 Therefore, on Steady you find quizzes and  the pdf versions for all the videos here. 

8 00:00:35,520 –> 00:00:41,840 Okay, now you might know I already have  some calculation videos in linear algebra. 

9 00:00:41,840 –> 00:00:45,680 But with this course here, you  finally get the whole structure  

10 00:00:45,680 –> 00:00:50,480 and should be able to learn linear algebra, by watching the videos in the right order. 

11 00:00:51,440 –> 00:00:56,640 Indeed, linear algebra as a whole is  a very important topic in mathematics, 

12 00:00:56,640 –> 00:00:58,880 and it has a lot of different applications. 

13 00:01:00,080 –> 00:01:04,400 One application, I can visualise  immediately with a short drawing here. 

14 00:01:05,360 –> 00:01:10,000 Just imagine in the problem you want to  solve, a lot of variables are involved. 

15 00:01:10,880 –> 00:01:14,720 For example, we could have some  fixed quantities, we just call  

16 00:01:14,720 –> 00:01:18,160 a, b, c, d and so on. And on the other hand,  

17 00:01:18,160 –> 00:01:21,920 we also could have some variables where we don’t know the value yet. 

18 00:01:22,960 –> 00:01:26,720 Therefore, we call them unknowns, and here we denote them by x,y,z. 

19 00:01:28,640 –> 00:01:33,440 Hence, in order to solve your problem. You want to find possible values for x,y,z. 

20 00:01:35,040 –> 00:01:40,880 And exactly there, linear algebra can help when we know the relations between the quantities. 

21 00:01:41,760 –> 00:01:47,440 However, it turns out, we need simply  relations between our variables. 

22 00:01:48,640 –> 00:01:55,360 Then, when we have this, we can use linear algebra to find all possible solutions for the unknowns 

23 00:01:56,560 –> 00:02:02,240 Therefore, linear algebra is the important  tool which helps us solving our problem. 

24 00:02:03,200 –> 00:02:04,320 So, you could say: 

25 00:02:04,320 –> 00:02:09,040 Giving calculation rules for solutions  is one part of linear algebra. 

26 00:02:10,160 –> 00:02:16,000 Indeed, this is important. But in order to understand the topic in a whole, 

27 00:02:16,000 –> 00:02:21,120 we need to go abstract. Therefore, this abstract level is something, 

28 00:02:21,120 –> 00:02:26,560 we also will discuss in this course. For example, on this abstract level, 

29 00:02:26,560 –> 00:02:30,160 we will find a fundamental concept we call a vector space. 

30 00:02:31,200 –> 00:02:35,840 Then, in this space, we will  find objects, we call vectors. 

31 00:02:36,480 –> 00:02:40,480 And we will find out, we can  do a lot with these vectors, 

32 00:02:40,480 –> 00:02:44,160 which leads us to a new concept, we call a linear map. 

33 00:02:45,280 –> 00:02:49,840 Hence, there we have now something  that can transform vectors. 

34 00:02:50,720 –> 00:02:55,520 Now, please don’t be deterred here because we don’t start at the abstract level. 

35 00:02:56,400 –> 00:02:59,120 First, we will consider we will  consider a very concrete level. 

36 00:03:00,080 –> 00:03:06,000 This means, instead of an abstract vector space, we start with something, we can call Rn. 

37 00:03:07,440 –> 00:03:10,720 Then, on this level, a linear  map just corresponds to 

38 00:03:10,720 –> 00:03:16,320 something we can put into a table. And such a table, we just call a matrix. 

39 00:03:17,200 –> 00:03:19,440 Okay, now we now: In this course,  

40 00:03:19,440 –> 00:03:24,640 we first discuss the concrete level here and then we go to the abstract level afterwards. 

41 00:03:25,680 –> 00:03:28,640 Of course, there are some  prerequisites for this course 

42 00:03:28,640 –> 00:03:29,280 you should know. 

43 00:03:30,240 –> 00:03:33,520 However, the good thing is,  I have a whole series about 

44 00:03:33,520 –> 00:03:36,480 start learning mathematics  that should help you there. 

45 00:03:37,600 –> 00:03:42,400 Now, you don’t have know all the topics there but it would be helpful if you  

46 00:03:42,400 –> 00:03:46,960 gather a basic knowledge about logical symbols, set operations and maps. 

47 00:03:47,920 –> 00:03:51,760 In particular, working with maps,  will be important immediately. 

48 00:03:52,880 –> 00:03:55,680 Then, with this knowledge,  I would say, you are ready 

49 00:03:55,680 –> 00:04:00,400 to start with linear algebra here. Therefore, in the next video,  

50 00:04:00,400 –> 00:04:04,240 we can immediately talk about vectors and how to calculate with them. 

51 00:04:05,120 –> 00:04:13,840 So I really hope that I see you there. Have a nice day and bye!

Q1: Which of these set relations is false?

A1: $1 \in \mathbb{R}$.

A2: $\sqrt{2} \in \mathbb{R}$.

A3: $\frac{5}{9} \in \mathbb{R}$.

A4: $0.\overline{9} \in \mathbb{R}$.

A5: $\pi \in \mathbb{R}$

A6: $-5 \in \mathbb{R}$

A7: None of them.

Q2: Which of these statements for the absolute value $|\cdot|$ and real numbers $x,y \in \mathbb{R}$ is not correct?

A1: $|x + y| = |x| + |y|$.

A2: $|x \cdot y| = |x| \cdot |y|$.

A3: $|x + y| \leq |x| + |y|$.

A4: $|x \cdot y| \leq |x| \cdot |y|$.

Q3: Is the map $f: \mathbb{R} \rightarrow \mathbb{R}$ given by $x \mapsto x^2$ injective?

A1: Yes!

A2: No!

Q4: Is the map $f: [0,\infty) \rightarrow \mathbb{R}$ given by $x \mapsto x^2$ injective?

A1: Yes!

A2: No!