Shadowing Practice: Essence of linear algebra preview - Learn English Speaking with Video
Creating lesson...
1
Hey everyone!
2
So, I'm pretty excited about the next sequence of videos that I'm doing.
3
They'll be about linear algebra, which as a lot of you know, is one of those subjects that's required knowledge for just about any technical discipline, but it's also, I've noticed,
4
generally poorly understood by students taking it for the first time time.
5
A student might go through a class and learn how to compute lots of things, like matrix multiplication, or the determinant, or cross products which use the determinant,
6
or eigenvalues, but they might come out without really understanding why matrix multiplication is defined the way that it is,
7
why the cross product has anything to do with the determinant, or what an eigenvalue really represents.
8
Oftentimes, students end up well-practiced in the numerical operations of matrices, but are only vaguely aware of the geometric intuitions underlying it all.
9
But there's a fundamental difference between understanding linear algebra on a numeric level and understanding it on a geometric level.
10
Each has its place, but roughly speaking, the geometric understanding is what lets you judge what tools to use to solve specific problems,
11
feel why they work, and know how to interpret the results.
12
And the numeric understanding is what lets you actually carry through the application of those tools.
13
Now, if you learn linear algebra without getting a solid foundation in that geometric understanding, the problems can go unnoticed for a while,
14
until you've gone deeper into whatever field you happen to pursue, whether that's computer science, engineering, statistics, economics, or even math itself.
15
Once you're in a class, or a job for that matter, that assumes fluency with linear algebra,
16
the way that your professors or your co-workers apply that field could seem like utter magic.
17
They'll very quickly know what the right tool to use is, and what the answer roughly looks like in a way
18
that would seem like computational wizardry if you assume that they're actually crunching all the numbers in their head.
19
Here, as an analogy, imagine that when you first learned about the sine function in trigonometry, you were shown this infinite polynomial.
20
This, by the way, is how your calculator evaluates the sine function.
21
For homework, you might be asked to practice computing approximations of the sine function, by plugging in various numbers to the formula and cutting it off at a reasonable point.
22
And in fairness, let's say you had a vague idea that this was supposed to be related to triangles, But exactly how had never really been clear, and was just not the focus of the course.
23
Later on, if you took a physics course, where sines and cosines are thrown around left and right, and people are able to tell pretty immediately how to apply them
24
and roughly what the sign of a certain value will be, it would be pretty intimidating, wouldn't it?
25
It would make it seem like the only people who are cut out for physics are those with computers for brains, and you would feel unduly slow or dumb for taking so long on each problem.
26
It's not that different with linear algebra.
27
And luckily, just as with trigonometry, there are a handful of intuitions, visual intuitions, underlying much of the subject.
28
And unlike the trig example, the connection between the computation and these visual intuitions is typically pretty straightforward.
29
And when you digest these, and really understand the relationship between the geometry and the numbers, the details of the subject, as well as how it's used in practice,
30
start to feel a lot more reasonable.
31
In fairness, most professors do make an effort to convey that geometric understanding.
32
The science example is a little extreme.
33
But I do think
34
that a lot of courses have students spending a disproportionate amount of time on the numerical side of things, especially given that in this day and age,
35
we almost always get computers to handle that half, while in practice, humans worry about the conceptual half.
36
So this brings me to the upcoming videos.
37
The goal is to create a short, binge-watchable series animating those intuitions from the basics of vectors up through the core topics
38
that make up the essence of linear algebra.
39
I'll put out one video per day for the next five days, then after that put out a new chapter every one to two weeks.
40
I think it should go without saying that you cannot learn a full subject with a short series of videos, and that's just not the goal here.
41
But what you can do, especially with this subject, is lay down all the right intuitions
42
so the learning that you do moving forward is as productive and fruitful as it can be.
43
I also hope this can be a resource for educators who are teaching courses that assume fluency with linear algebra, giving them a place to direct students that need a quick brush up.
44
I'll do what I can to keep things well paced throughout, but it's hard to simultaneously account for different people's different backgrounds and levels of comfort,
45
so I do encourage you to readily pause and ponder if you feel that it's necessary.
46
Actually, I'd give that same advice for watching any math video, even if it doesn't feel too quick, since the thinking that you do on your own time is where all the learning really happens, don't you think?
47
So, with that as an introduction, I'll see you next video.
📺 Same Channel
C11
Linear combinations, span, and basis vectors | Chapter 2, Essence of linear algebra
American AccentCambridge Exam PrepUniversity StudentsWorking Adults
C12
Euler's Formula Poem
American AccentIELTS PrepTOEFL PrepUniversity Students
C14
The inscribed square problem
American AccentUniversity StudentsWorking Adults
C13
The Newton art puzzle
American AccentUniversity StudentsWorking Adults
C12
How might LLMs store facts | Deep Learning Chapter 7
American AccentIELTS PrepTOEFL PrepUniversity Students
C15
Transformers, the tech behind LLMs | Deep Learning Chapter 5
American AccentBusiness EnglishUniversity StudentsWorking Adults
C15
Fourier series
American AccentUniversity StudentsWorking Adults
C21
Hilbert's Curve: Is infinite math useful?
American AccentBusiness EnglishUniversity StudentsWorking Adults
C12
Vectors | Chapter 1, Essence of linear algebra
American AccentBusiness EnglishUniversity StudentsWorking Adults
C237
Large Language Models explained briefly
American AccentBusiness EnglishUniversity StudentsWorking Adults
C23
But what is a neural network? | Deep learning chapter 1
American AccentBusiness EnglishUniversity StudentsWorking Adults
C12
Oh, wait, actually the best Wordle opener is not “crane”…
American AccentBusiness EnglishUniversity StudentsWorking Adults
C1Divergence and curl: The language of Maxwell's equations, fluid flow, and more
American AccentBusiness EnglishUniversity StudentsWorking Adults
C1But what is quantum computing? (Grover's Algorithm)
American AccentBusiness EnglishUniversity StudentsWorking Adults
C1Tie random ends: How many loops?
American AccentUniversity StudentsWorking Adults
✨ Recommend Video
C186 sentences
The End of Human Mathematics? AI's Recent Breakthrough -- And What's Next
Business EnglishUniversity StudentsWorking Adults
C12
14 sentences
CS50x 2026 - Introduction
Business EnglishUniversity StudentsWorking Adults
C11
184 sentences
PLC Programming Tutorial - Ladder Logic #3: Math and Calculations
Business EnglishUniversity StudentsWorking Adults
C17
175 sentences
The Lost Art Of The Intellectual Life
Business EnglishUniversity StudentsWorking Adults
B22
261 sentences
Your Life as Every Industrial Engineer Rank
Business EnglishUniversity StudentsWorking Adults
C12
55 sentences
Convergence and Divergence: The Return of Sequences and Series
American AccentCambridge Exam PrepUniversity StudentsWorking Adults
C19
38 sentences
The Neuroscience of Learning
Business EnglishUniversity StudentsWorking Adults
C11
120 sentences
Learn Data Modeling in 8 minutes: Dimensional Data Modeling, Data Vault, and One Big Table
Business EnglishUniversity StudentsWorking Adults
C11
126 sentences
How I use AI as a senior software engineer and 5 principles of AI explained in 10 minutes
Business EnglishUniversity StudentsWorking Adults
C11
108 sentences
How I Learned Procedural Generation
Business EnglishUniversity StudentsWorking Adults
C111
93 sentences
Naive Bayes
Business EnglishUniversity StudentsWorking Adults
Vocabulary and speaking notes for this lesson
This C1 speaking lesson is built on the video “Essence of linear algebra preview”. The speaker keeps coming back to these words: algebra, linear, understanding, intuitions, geometric. This video has 47 sentences and 917 words to shadow. The speech runs for 4:43. The speaker talks fast, about 195 words per minute, so expect linked and reduced sounds. Only 81% of the words are among the 3,000 most common in English, so the vocabulary is demanding.
Key vocabulary in this video
The 15 most advanced words in the video, with pronunciation and meaning:
| Word | Pronunciation | Meaning |
|---|---|---|
| algebra noun | /ˈæl.d͡ʒɪ.bɹə/ | Elementary algebra: A system for representing and manipulating unknown quantities (variables) in equations. |
| intuition noun | /ˌɪn.tjuːˈɪʃ.ən/ | Immediate cognition without the use of conscious rational processes. |
| geometric adjective | /ˌd͡ʒiː.əʊˈmɛt.ɹɪk/ | Of or relating to geometry. |
| sine noun | /saɪn/ | In a right triangle, the ratio of the length of the side opposite an angle to the length of the hypotenuse. |
| determinant noun | /dɪˈtɜːmɪnənt/ | A determining factor; an element that determines the nature of something. |
| matrix noun | /ˈmeɪ.tɹɪks/ | The cavity or mold in which anything is formed. |
| compute verb | /kəmˈpjuːt/ | To reckon, calculate. |
| eigenvalue noun | /ˈaɪ.ɡənˌvæl.ju/ | A scalar λ, such that there exists a non-zero vector x (a corresponding eigenvector) for which the image of x under a given linear transformation A is equal to… |
| fluency noun | /ˈfluːənsi/ | The quality of smoothness of flow. |
| multiplication noun | /ˌmʌltɪplɪˈkeɪʃən/ | The process of computing the sum of an addition with one and the same number a specified number of times (i.e. 4 times 5 = [0 +] 5 + 5 + 5 + 5 = 20) or any… |
| numeric adjective | /n(j)uːˈmɛɹɪk/ | Of or relating to numbers, especially the characters 0 to 9. |
| trigonometry noun | /ˌtɹɪɡəˈnɒmətɹi/ | The branch of mathematics that deals with the relationships between the sides and angles of (in particular) right-angled triangles, as represented by the… |
| numerical adjective | /n(j)uːˈmɛɹ.ɪ.kəl/ | Of or pertaining to numbers. |
| fairness noun | /ˈfɛɚ.nəs/ | The property of being fair or equitable. |
| animate adjective | /ˈæn.ə.mət/ | That lives. |
Phrasal verbs you will hear
| Word | Pronunciation | Meaning |
|---|---|---|
| put out verb | To blind (eyes). | |
| brush up verb | To paint, or make clean or bright with a brush. | |
| carry through verb | To manage to execute; to perform successfully, all the way to the end. | |
| come out verb | /ˌkʌm ˈaʊt/ | To be discovered; to be revealed. |
| end up verb | To bring to a conclusion. | |
| go through verb | To travel from one end of something to the other. | |
| lie down verb | To assume a horizontal position. | |
| make up verb | /ˌmeɪk ˈʌp/ | To constitute, to compose. |
Grammar in this video
The structures the speaker uses most, with the exact words from the video:
| Structure | In the video |
|---|---|
| Passive voice be + past participle — the focus is on what happens, not who does it | is defined · were shown · might be asked |
| Present perfect have/has + past participle — a past action that still matters now | I've noticed · you've gone |
Pronunciation to watch
The speaker uses 15 contractions and reduced forms, such as I'll, I'm, they'll. Say them the short way, as you hear them.
- The “sh” and “zh” sounds: intuition /ˌɪn.tjuːˈɪʃ.ən/, multiplication /ˌmʌltɪplɪˈkeɪʃən/, approximation /əˌpɹɑk.sɪˈmeɪ.ʃən/, computation /ˌkɑmpjʊˈteɪʃn̩/, disproportionate /dɪs.pɹəˈpɔː.ʃə.nət/
- Long words — get the stress right: intuition /ˌɪn.tjuːˈɪʃ.ən/, geometric /ˌd͡ʒiː.əʊˈmɛt.ɹɪk/, determinant /dɪˈtɜːmɪnənt/, eigenvalue /ˈaɪ.ɡənˌvæl.ju/, multiplication /ˌmʌltɪplɪˈkeɪʃən/
How to practise with this video
- Listen to the whole video once without speaking and note the words you do not know.
- Start at 0.75× speed, shadow it sentence by sentence, then go back to normal speed once it feels easy.
- Record yourself and compare with the original, paying attention to words like algebra, intuition, geometric.
What is the Shadowing Technique?
Shadowing is a science-backed language learning technique originally developed for professional interpreter training and popularized by polyglot Dr. Alexander Arguelles. The method is simple but powerful: you listen to native English audio and immediately repeat it out loud — like a shadow following the speaker with just a 1–2 second delay. Unlike passive listening or grammar drills, shadowing forces your brain and mouth muscles to simultaneously process and reproduce real speech patterns. Research shows it significantly improves pronunciation accuracy, intonation, rhythm, connected speech, listening comprehension, and speaking fluency — making it one of the most effective methods for IELTS Speaking preparation and real-world English communication.