• Title: Introduction

  • Series: Linear Algebra

  • Chapter: Vectors in $ \mathbb{R}^n $

  • YouTube-Title: Linear Algebra 1 | Introduction

  • Bright video: https://youtu.be/x2cYoSPGz3o

  • Dark video: https://youtu.be/SrhtsGd0y1s

  • Quiz: Test your knowledge

  • PDF: Download PDF version of the bright video

  • Dark-PDF: Download PDF version of the dark video

  • Print-PDF: Download printable PDF version

  • Thumbnail (bright): Download PNG

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  • Subtitle on GitHub: la01_sub_eng.srt

  • Timestamps

    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

  • Subtitle in English

    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!

  • Quiz Content

    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!

  • Last update: 2024-10

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