Shadowing Practice: Give Me 30 min, I will make Linear Algebra Click Forever - Learn English Speaking with Video
Choose a practice mode
Each mode opens a full-screen practice room.
560 sentences
If sentences are too short or too long, click Edit to adjust them.
1
Give me 30 minutes.
2
Just 30 minutes.
3
And I will make linear algebra click.
4
Forever.
5
You've probably seen these scary words before.
6
Dot product, Gaussian elimination, vector space, determinants.
7
They look like a foreign language.
8
Maybe from some textbook you were forced to read and immediately forgot.
9
But this isn't going to be some dry lecture.
10
We're going to build your intuition from the ground up.
11
And the best part?
12
we're going to show you how every single one of these concepts connects.
13
Okay, take a look at this table on the screen.
14
This is our entire journey.
15
It's our roadmap.
16
Take a look at this table.
17
We'll start with vectors and the dot product, measuring similarity.
18
Then linear systems, ax equals b, solving complex problems.
19
Vector spaces, span, basis, the rules of your world.
20
And linear transformations, where matrices become actions.
21
plus determinants.
22
One number that reveals everything.
23
I get it.
24
Right now, looking at all of this, it might look intimidating.
25
But here is my promise to you.
26
By the end of this video, every single row you see on that screen will make perfect intuitive sense.
27
Now it's time to tackle that first row.
28
We're starting with the most fundamental object in this entire field.
29
The vector.
30
But forget arrows in a physics textbook.
31
That's not what we're talking about today.
32
Let's ask a real question.
33
How does a streaming service know you'll like a movie?
34
How can it possibly quantify something as fuzzy as personal taste?
35
It all starts with representing your taste as a vector.
36
Let's look at the movie ratings example.
37
Imagine three users and their ratings for three movies.
38
We've got sci -fi, comedy, and romance.
39
User A gives us the numbers 5, 4, 1.
40
They love sci -fi, they like comedy, but they hate romance.
41
User B gives us 4, 5, 2.
42
Pretty similar to user A.
43
And user C gives us 1, 2, 5.
44
Very different.
45
Right now, these are just lists of numbers.
46
But if we treat them as coordinates in a 3D movie space, something magical happens.
47
Forget the math for a second.
48
Seriously, just look at this picture.
49
You can see the user taste in this 3D movie space.
50
Here is the key idea.
51
The smaller the angle between two vectors, the more similar they are.
52
That's it.
53
That's the core intuition.
54
Our entire problem now is just to calculate that angle.
55
The tool we use is called the dot product.
56
Now, the dot product has two formulas but one goal.
57
Here's the secret.
58
We use the easy one, the calculation one, to solve for the hard one, the geometry one.
59
First, the calculation formula.
60
This is easy.
61
You just multiply corresponding components and add them up.
62
a .b equals a1b1 plus a2b2 plus a3b3.
63
Simple.
64
Second, the geometric formula.
65
This connects the dot product to the angle, theta.
66
a .b equals the length of a times the length of b times the cosine of theta.
67
And that length, by the way, is just the square root of a1 squared plus a2 squared plus a3 squared.
68
So here's the game plan.
69
1. We calculate A .B using the easy calculation formula.
70
2. We calculate the lengths of A and B. And 3.
71
We plug those numbers into the geometric formula to solve for the angle theta.
72
Let's do it.
73
Our user vectors are A equals 5, 4, 1, B equals 4, 5, 2, and C equals 1, 2, 5.
74
Case 1.
75
Comparing similar users A and B.
76
Step 1.
77
Calculate the dot product A .B.
78
We take 5 times 4 plus 4 times 5 plus 1 times 2, which equals 42.
79
Okay, step 2.
80
Calculate the vector lengths.
81
The length of A is the square root of 5 squared plus 4 squared plus 1 squared.
82
That's the square root of 25 plus 16 plus 1, which is the square root of 42.
83
About 6 .48.
84
The length of B is the square root of 4 squared plus 5 squared plus 2 squared.
85
That's 16 plus 25 plus 4, square root of 45.
86
About 6 .71.
87
Almost there.
88
Step 3.
89
Solve for the angle theta.
90
We rearrange the formula.
91
Cosine theta equals a dot b divided by the length of a times the length of b.
92
So that's 42 divided by 6 .48 times 6 .71, which gives us about 0 .966.
93
So theta is the arc cosine of 0 .966, which is approximately 15 degrees.
94
That's tiny.
95
The math just proved what our eyes told us.
96
These users have very similar tastes.
97
Okay, case two.
98
Comparing dissimilar users A and C.
99
Quick quiz.
100
What do you expect?
101
A bigger angle or a smaller one?
102
Let's find out.
103
Our user vectors are a, which is 5, 4, 1, and c, which is 1, 2, 5.
104
Step 1, the dot product, a .c, equals 18.
105
Step 2, the length of c, that's the square root of 1 squared plus 2 squared plus 5 squared, which is about 5 .48.
106
Step 3.
107
Solve for the angle.
108
Cosine theta equals 18 divided by 6 .48 times 5 .48.
109
That comes out to about 0 .507.
110
So theta is the arc cosine of that, which is 59 .5 degrees.
111
A much wider angle.
112
The number 59 .5 degrees is our precise mathematical measure of how different their tastes are.
113
You just understood the entire theory.
114
Now, let's get some instant gratification with code.
115
We import numpy, we define our user ratings.
116
A is a numpy array of 5, 4, 1.
117
B is 4, 5, 2.
118
And C is 1, 2, 5.
119
Now the magic is in these two lines.
120
Watch this.
121
The dot product for A and B is just np .dot with A and B.
122
Then the cosine of theta is
123
that dot product divided by the norm of A times the norm of B And then we just convert that to degrees
124
We do the same thing for A and C And when we print it out
125
The angle between A and B is 15 .0 degrees
126
The angle between A and C is 59 .5 degrees
127
The exact same results
128
You've seen the intuition
129
You've done the math
130
You've seen the code
131
The dot product is a machine for measuring similarity.
132
So, we've handled single vectors.
133
But what happens when we need to solve problems with systems of relationships all at once?
134
This is where linear algebra was born.
135
It wasn't born from abstract math, but from concrete problems.
136
Let's look at one.
137
The problem.
138
Coffee blends.
139
A coffee shop has two special blends and a limited stock of beans.
140
Okay, so blend A requires 300 grams of Arabica and 100 grams of Robusta.
141
Got it.
142
Blend B is a bit different.
143
It needs 100 grams of Arabica and 200 grams of Robusta.
144
You look at your total stock, and you have 11 ,000 grams of Arabica and 8 ,000 grams of Robusta.
145
So here's the question.
146
How many units of each blend can you make to use up your entire stock?
147
This is a classic linear algebra problem.
148
Now, let's think about the high school way of doing this.
149
If we make x units of blend A and y units of blend B, we'd write it out.
150
For Arabica, it's 300x plus 100y equals 11 ,000.
151
And for Robusta, it's 100x plus 200y equals 8 ,000.
152
This is messy.
153
Writing out equations like this is slow, and, frankly, it doesn't scale well.
154
Linear algebra gives us a clean, powerful way to organize all of this.
155
The blueprint AX equals B.
156
Now, don't panic.
157
This isn't a new rule you have to memorize.
158
It's just a filing system, a way to keep things tidy.
159
Let's look at the pieces.
160
First, we have A.
161
This is the recipes matrix.
162
See here?
163
The first column is the recipe for blend A, 300 and 100.
164
The second column is for blend B, 100 and 200.
165
Next is X.
166
That's the what you want vector.
167
It holds our unknowns, x and y.
168
And finally, b.
169
That's our stock vector.
170
The total stuff we have, 11 ,000 and 8 ,000.
171
Putting it all together, our entire problem is now just one clean line.
172
The matrix with 300, 100, 100, 200 times the vector xy equals the vector 11 ,000, 8 ,000.
173
This is the universal language for systems of linear equations.
174
So how do we solve it?
175
The algorithm is a game!
176
It's called Gaussian elimination!
177
Here's the game plan.
178
The goal.
179
Turn our matrix A into an upper triangle shape.
180
This is called row echelon form.
181
And the legal moves?
182
There are only three.
183
One, you can swap any two rows.
184
Two, you can multiply an entire row by a non -zero number.
185
And three, you can add a multiple of one row to another row.
186
That's it.
187
Let's play.
188
Step one.
189
Set up the game board.
190
We combine A and B into what's called an augmented matrix.
191
It looks like this.
192
A 2x3 grid with 300, 100, and 11 ,000 on the top row and 100, 200, and 8 ,000 on the bottom.
193
Step 2.
194
Simplify the board.
195
We can use rule 2.
196
These numbers are huge.
197
Let's divide the first row, R1, by 100, and let's divide the second row, R2, by 102.
198
Look at that!
199
We go from that big messy board to a new one.
200
3 -1, 110 on top, and 1 -2, 80 on the bottom.
201
Much cleaner.
202
Step 3.
203
Get a 1 in the top left pivot.
204
We'll use rule 1 for this.
205
It's always easier to work with a 1.
206
So, let's just swap row 1 and row 2.
207
And now, our board looks like this.
208
1 -2, 80 on top, and 3 -1, 110 on the bottom.
209
Perfect.
210
Our first pivot is locked in.
211
Step 4.
212
Eliminate below the pivot.
213
This is the key move using rule 3.
214
We want to turn that 3 in the bottom left into a 0.
215
So we'll make a new row 2 by taking the old row 2 and subtracting 3 times row 1 from it.
216
Let's do the math.
217
For the first element, 3 minus 3 times 1 is 0.
218
For the second, 1 minus 3 times 1 is negative 2.
219
2 is negative 5.
220
And for the last one, 110 minus 3 times 80, which is negative 130.
221
And we're done.
222
We've won the game.
223
Look at that beautiful upper triangle shape.
224
This is row echelon form.
225
Step 5.
226
The payoff.
227
It's called back substitution.
228
We just turn this back into equations.
229
The bottom row tells us 0x minus 5y equals negative 130.
230
That's easy.
231
Y equals negative 130 divided by negative 5, which is 26.
232
Now we take the top row.
233
1x plus 2y equals 80.
234
We know y is 26, so we plug it in.
235
x plus 2 times 26 equals 80, which means x is 28.
236
The result?
237
The shop can make 28 units of blend A and 26 units of blend B.
238
You just solved a real -world system from scratch.
239
You just did the hard work to build the intuition, and what's your reward?
240
In the real world, you'll never do that by hand again.
241
Your reward is the one -liner.
242
We import numpy as np, we define our recipe matrix A as a numpy array with our values,
243
we define our stock vector B, and then the magic.
244
x equals np .lin alg .solve with A and B inside.
245
We print the results, and...
246
units of blend A, 28 .0, units of blend B, 26 .0.
247
And now you know exactly what that one line of code is doing under the hood.
248
So, we've solved a specific problem with a specific amount of beans.
249
But what are the rules of this coffee -making world?
250
What are all the possible combinations of blends we could have made?
251
To answer that, we need to define the rules of the world our vectors live in.
252
Now, I'll be honest, this chapter is the most abstract one we're going to cover.
253
But I'm going to make it click with a simple analogy.
254
Forget coffee.
255
You're programming a 2D video game character.
256
You give your character two move commands, move A and move B.
257
Here are the two critical questions.
258
First, can your character reach every single point on the screen with just those two moves?
259
And second, are both moves actually necessary, or is one of them redundant?
260
Answering these two questions is the entire point of this chapter.
261
Let's build the core vocabulary of your world.
262
The standard 2D grid uses two building block vectors to get anywhere.
263
The first is I, which is 1, 0.
264
That's just one step to the right.
265
The second is J, which is 0, 1, or one step up.
266
So to get to the point 3, 4, you just take a scaled sum.
267
You take 3 times the right vector, plus 4 times the up vector, and you land on 3, 4.
268
This is called a linear combination.
269
Our game characters' moves are just different building blocks.
270
So let's define our terms.
271
First, span.
272
This answers the question, where can my character go?
273
The span is the set of all possible points you can reach.
274
Does it cover just a tiny line, or does it cover the entire 2D plane?
275
Second term, linear independence.
276
This answers the question, are any of my moves redundant?
277
If move B is just two times move A, they are linearly dependent.
278
The second move is useless.
279
It doesn't unlock any new directions.
280
You're just stuck on the same path.
281
Which brings us to our third term, basis.
282
This is the perfect set of moves.
283
A basis has to be two things.
284
First, it's linearly independent, so there are no redundant moves.
285
And second, it spans the entire space, so you can reach everywhere.
286
A basis is the minimal, most efficient set of building blocks for your world.
287
For our 2D game screen, any basis must have exactly two vectors.
288
We say the dimension of the space is 2.
289
Let's look at some movesets.
290
You'll see the difference instantly.
291
Okay, in this table we're going to look at four different sets of moves.
292
We'll ask if they span the plane, if they're independent, and if they make a basis.
293
Case 1, the standard basis.
294
The vectors are 1, 0, and 0, 1.
295
Does it span the 2D plane?
296
Yes.
297
Can it get everywhere?
298
Is it linearly independent?
299
Yes.
300
No redundancy.
301
So, is it a basis?
302
Yes.
303
It's perfectly efficient.
304
Case 2.
305
The set with vectors 1, 1, and 2, 2.
306
Does it span the plane?
307
No. You're totally stuck on a line.
308
Are they independent?
309
The second move is just twice the first.
310
It's totally redundant.
311
It adds nothing new.
312
So it's definitely not a basis.
313
What if we have three moves?
314
1, 0, 0, 1, and 1, 1.
315
Does it span the plane?
316
Yes, you can certainly get everywhere.
317
But is it independent?
318
The third vector is just the first two added together.
319
It's unnecessary.
320
So it's not a basis because it's not efficient.
321
It's bloated.
322
Case 4, a non -standard basis.
323
The vectors are 1, 2, and 3, 1.
324
Does it span the plane?
325
Yes, they point in different enough directions.
326
Are they independent?
327
Yes, you can't make one from the other.
328
So is it a basis?
329
Yes, it's a perfect, efficient, and complete set of moves, even though they're a bit skewed.
330
Okay, you're thinking, I get the game analogy, But why should I care about a non -standard basis?
331
Because changing your basis is a superpower.
332
It lets you describe the same data from a more effective point of view.
333
It's the core idea behind JPEG image compression.
334
It's also the core idea behind
335
Principal Component Analysis, or PCA.
336
Let's connect this back to what you already know.
337
Here's the problem.
338
We have a point in the standard basis, P equals 7, 5.
339
And we have our new skewed basis, b, made of the vectors 1, 2, and 3, 1.
340
The question is, what are p's coordinates in our new basis?
341
We're looking for two numbers.
342
Let's call them c1 and c2, such that c1 times our first basis vector plus c2 times our second basis vector equals our point p.
343
Does this look familiar?
344
It's the exact same ax equals b problem from the last chapter.
345
We can write it as a matrix equation.
346
A matrix with our basis vectors as columns times a vector with our unknown coordinates c1
347
and c2 equals our point vector p.
348
We're just solving bc equals p.
349
The concepts are connecting.
350
Let's solve it.
351
We import numpy as in p.
352
Our new basis vectors form the columns of a matrix we'll call b.
353
So b equals in p dot array with rows 1, 3, and 2, 1.
354
The point p we want to express is an array with 7, 5.
355
Then we solve the system BC equals P for C, which are the new coordinates.
356
So C equals inP dot lin alg dot solve, passing in B and P.
357
Let's print the coordinates of P in our new basis B.
358
The output is 1 .6 and 1 .8.
359
This means we've described the exact same point using a different set of building blocks.
360
Our point P is just 1 .6 times the vector 1, 2, plus 1 .8 times the vector 3, 1.
361
This isn't just a party trick, it's a foundational technique in modern data analysis and computer graphics.
362
We just established the quote -unquote rules of the world our vectors live in.
363
We talked about basis vectors and how they define our grid.
364
But that world was static.
365
It just sat there.
366
Now, we're going to make things happen.
367
And here is the single biggest mindset shift in all of linear algebra.
368
Up until now, you've probably seen matrices as boring grids of numbers, something you use for solving A times X equals B.
369
Forget that.
370
Erase it from your mind.
371
From this moment on, a matrix is an action.
372
It's a verb.
373
It rotates, scales, and shears space itself.
374
The equation V' equals MV doesn't mean solve for X
375
it means the matrix M acts on the vector V to produce a new transformed vector, V'.
376
Let's make this real with a problem, rotating a game character.
377
Imagine you've got this triangle on the screen.
378
Its points before the rotation are P1 equals 11, P2 is 31, and P3 is 22.
379
Your job is to rotate this triangle 90 degrees counterclockwise around the origin.
380
How do you find the new coordinates?
381
The after coordinates you want are P1' at minus 1, 1, P2' at minus 1, 3, and P3' at minus 2, 2.
382
But how do you get there systematically?
383
Here's the secret.
384
The key to understanding any linear transformation.
385
You only need to watch where the basis vectors i and j land.
386
That's it.
387
That's the whole trick.
388
Let's watch it happen for our 90 degree rotation.
389
Before the transformation, our basis vectors are the usual suspects.
390
i is the vector pointing right, and j is the vector pointing straight up.
391
Now, we apply the 90 degree rotation.
392
What happens?
393
i, which was at gets rotated and now it lands at .
394
And J, which was at gets rotated and lands over at . And here's the magic.
395
The columns of a transformation matrix are just the coordinates of where the original basis vectors land.
396
So let's build our matrix.
397
1. The first column is where I landed.
398
That's the vector .
399
2. The second column is where J landed.
400
That's the vector . And BOOM!
401
Looking at this matrix R, we have a first column of 0, 1, and a second column of minus 1, 0.
402
This is the genetic code for a 90 degree rotation.
403
Now for the payoff, let's rotate that triangle.
404
Here's the setup.
405
We have our rotation matrix R, and we have our three vertices, P1, P2, and P3.
406
Step 1.
407
Let's transform vertex P1.
408
We calculate P1' equals R times P1, so that's our rotation matrix, times the vector 1 ,1.
409
The result is the vector minus 1 ,1.
410
So our new point P1' is at minus 1 ,1.
411
Exactly what we wanted.
412
Step 2.
413
Transform vertex P2.
414
Same deal.
415
P2' is R times P2, our matrix, times the vector 3 ,1.
416
And that gives us the vector minus 1 ,3.
417
The new point P2' is at .
418
Step 3.
419
And finally, transform vertex P3.
420
P3' is R times P3.
421
That's our matrix times the vector .
422
The result is .
423
The new point P3' is at .
424
Look at that!
425
We have the before triangle and the after triangle.
426
A perfect rotation!
427
Now, from one triangle to millions.
428
This isn't some abstract concept.
429
It is the engine of every video game you have ever played.
430
Your graphics card, your GPU, is a piece of hardware optimized for one primary job.
431
Doing billions of v' equals mv multiplications per second.
432
So let's see how the professionals do it. With code.
433
First, the input.
434
We pack all our vertices into a single matrix.
435
We import numpy as np.
436
Here's our 90 -degree rotation matrix, R, which is a numpy array with rows 0, minus 1, and 1, 0.
437
Now here's the cool part.
438
Our triangle vertices become the columns of a single matrix, P.
439
Then, the calculation.
440
It's just one operation to transform all the points.
441
We write P underscore transformed equals R at P.
442
That at symbol is Python's way of saying matrix multiplication.
443
And the output?
444
Perfect results.
445
Instantly.
446
We can print the original vertices and the transformed vertices.
447
And look, the original matrix P has columns 1 1, 3 .1, and 2 .2.
448
The transformed matrix has columns minus 1 1, minus 1 3, and minus 2 2.
449
It's the exact same result we got by doing it one by one, but it's way more efficient.
450
This is the power of thinking of matrices as actions.
451
We saw that matrices are not just grids of numbers.
452
They are verbs.
453
They are actions that rotate, scale, and shear space.
454
We saw our rotation matrix R spin a triangle without changing its size at all.
455
But a scaling matrix?
456
That would have blown it up.
457
This leads to a critical question.
458
Isn't there one single number that just tells you how much a matrix scales area?
459
A single diagnostic number?
460
There is.
461
It's called the determinant.
462
Okay, forget the scary formulas for a moment.
463
Just for a moment.
464
Here is the single most important idea of this entire chapter.
465
The determinant is the area scaling factor of a transformation.
466
The determinant of a 2x2 matrix is the area of the parallelogram that the 1x1 unit square gets transformed into.
467
That's it.
468
That's the whole story.
469
As you can see in this little diagram, you start with a simple unit square with an area of 1.
470
You apply a transformation, M, and it becomes a parallelogram.
471
The area of that new parallelogram is the absolute value of the determinant of M.
472
This one number tells you everything.
473
For example, if the determinant of m is 1, then areas are perfectly preserved.
474
This is what you see in a pure rotation.
475
If the determinant of m is 2, the transformation doubles all areas.
476
And what if the determinant of m is less than 0, say, negative 1?
477
This means the orientation of space gets flipped.
478
It's like looking in a mirror.
479
And now, for the most important case of all.
480
If the determinant of m is zero, the area is zero.
481
The matrix has squished your entire world into a line, or maybe even just a single point.
482
You've lost a dimension, and the transformation is irreversible.
483
You can't go back.
484
This matrix is called singular, or non -invertible.
485
Okay, so let's see the algorithm.
486
For a 2x2 matrix, the calculation is actually super easy.
487
For a matrix A with elements A, B, C, D, the formula is right here.
488
The determinant of A equals A, D, minus B, C.
489
Alright, time for some rapid -fire diagnostics.
490
Let's analyze three transformations using just this one number.
491
First up, case 1, the 90 -degree rotation matrix.
492
The matrix R has rows 0, minus 1, and 1, 0.
493
Let's do the calculation.
494
The determinant of R is 0 times 0 minus minus 1 times 1.
495
So that's 0 minus negative 1, 1.
496
The result?
497
The determinant of R is 1.
498
The math confirms it.
499
Pure rotation preserves area perfectly.
500
You can see it here.
501
The unit square just spins, but its area stays exactly 1.
502
Next, case 2, a scaling matrix.
503
Our matrix S has rows 2, 0, and 0, 3.
504
The calculation is 2 times 3 minus 0 times 0.
505
That's 6 minus 0.
506
The result?
507
The determinant of S is 6.
508
This means this transformation multiplies all areas by exactly 6.
509
The unit square, which started with an area of 1, becomes a rectangle with an area of 6.
510
Area multiplied by 6.
511
And finally, case 3, the broken singular matrix.
512
Look at this matrix C.
513
The determinant of C is 0.
514
This is our red flag.
515
This matrix collapses 2D space onto a line.
516
As you see here, the unit square gets completely squashed.
517
It has an area of 0.
518
Space collapsed.
519
So here's your diagnostic toolkit.
520
These properties are incredibly useful.
521
If the determinant of A is not 0, then A is invertible.
522
The transformation can be reversed.
523
If the determinant of A is 0, then A is singular.
524
Space is squashed into a lower dimension.
525
Here's a cool one.
526
The determinant of A times B is just the determinant of A times the determinant of B.
527
So the scaling factors just multiply together.
528
And that means the determinant of A inverse is just 1 over the determinant of A.
529
It just reverses the scaling.
530
Remember our coffee problem from earlier?
531
ax equals b?
532
Before you even try to solve it, you can calculate the determinant of a.
533
If it's zero, you know immediately the system is broken.
534
Of course, in the real world, you can do this with a one -second check with code.
535
Let's see it.
536
First, the input.
537
We import numpy as np.
538
We create our three arrays.
539
R for rotation, S for scaling, and C for the collapsing matrix.
540
Then, the calculation.
541
It's super simple.
542
We create a variable det underscore r and set it to np .linalg .det of r, and we do the same for s and c.
543
And the output?
544
The determinant of r is 1, the determinant of s is 6, and the determinant of c is 0.
545
The results match perfectly.
546
The determinant is your matrix's vital sign, giving you an instant health check on your transformations.
547
And just like that, you did it.
548
Let's look at the journey we took.
549
The analogy became the concept.
550
You went from movie ratings to the dot product, understanding it as similarity.
551
From delicious coffee blends to solving systems of equations with AX equals B.
552
You went from video game moves to understanding span and basis as the fundamental rules of the world.
553
And finally, you saw that matrices are verbs, the powerful engine behind all computer graphics.
554
But here's the most important part.
555
This isn't the end.
556
It's the foundation for everything that comes next.
557
Things like eigenvectors, PCA, SVD, the master algorithms of machine learning and data science.
558
And because of the work you just put in, you are now ready to understand them.
559
Truly.
560
Linear Algebra has finally clicked for you.
Download App
Everything you need to speak fluently
AI PronunciationScore every sentence
IPA PracticeMaster every sound
VocabularyBuild your word bank
Vocab GameLearn while playing
Same Channel
FOR YOU
Recommend Video
C14
72 sentences
Video 2: Shape
Daily ConversationUniversity Students
C11
712 sentences
Master AI in 60 Minutes with Vector Databases!
Daily ConversationUniversity Students
C12
261 sentences
Your Life as Every Industrial Engineer Rank
Daily ConversationUniversity Students
C17
107 sentences
Gaussian Naive Bayes, Clearly Explained!!!
Daily ConversationUniversity Students
C11
508 sentences
Einstein and the Theory of Relativity | HD |
Daily ConversationUniversity Students
C111
39 sentences
You Will Only Like Half of This Video
Daily ConversationUniversity StudentsWorking Adults
C12
186 sentences
Learn Java in 14 Minutes (seriously)
Daily ConversationUniversity Students
C13
119 sentences
Mathematics and Physics | Lesson 2 | Understanding the Connection
Daily ConversationUniversity Students
C11
108 sentences
How I Learned Procedural Generation
Daily ConversationUniversity StudentsWorking Adults
C11
747 sentences
MIT Introduction to Deep Learning | 6.S191
Daily ConversationUniversity Students
C11
433 sentences
Give Me 24 minutes, I'll Give You 1,000+ hours of Claude Knowledge
Daily ConversationUniversity StudentsWorking Adults
Vocabulary and speaking notes for this lesson
This speaking lesson is built on the video “Give Me 30 min, I will make Linear Algebra Click Forever”. The speaker keeps coming back to these words: vector, matrix, equals, determinant, basis. This video has 560 sentences and 4294 words to shadow. The speech runs for 29:56. The speaker talks at a steady 143 words per minute, a comfortable pace for shadowing. Only 74% of the words are among the 3,000 most common in English, so the vocabulary is demanding.
Key vocabulary in this video
15 words from the video worth learning, with pronunciation and meaning:
| Word | Pronunciation | Meaning |
|---|---|---|
| vector noun | /ˈvɛktɚ/ | A directed quantity, one with both magnitude and direction; the signed difference between two points. |
| matrix noun | /ˈmeɪ.tɹɪks/ | The cavity or mold in which anything is formed. |
| minus | /ˈmaɪ.nəs/ | Made less or reduced by (followed by an expression of number or quantity). |
| solve verb | /sɒlv/ | To find an answer or solution to a problem or question; to work out. |
| blend verb | /blɛnd/ | To mingle; to mix; to unite intimately; to pass or shade insensibly into each other. |
| rotation noun | /ɹoʊˈteɪʃən/ | The act of turning around a centre or an axis. |
| transformation noun | /ˌtɹæns.fɔɹˈmeɪ.ʃən/ | The act of transforming or the state of being transformed. |
| linear adjective | /ˈlɪn.i.əɹ/ | Having the form of a line; straight or roughly straight; following a direct course. |
| spin verb | /spɪn/ | To rotate, revolve, gyrate (usually quickly); to partially or completely rotate to face another direction. |
| angle noun | /ˈæŋ.ɡəl/ | A figure formed by two rays which start from a common point (a plane angle) or by three planes that intersect (a solid angle). |
| triangle noun | /ˈtɹaɪˌæŋɡəl/ | A polygon with three sides and three angles. |
| transform verb | /tɹænsˈfɔɹm/ | To change greatly the appearance or form of. |
| formula noun | /ˈfɔː.mjʊ.lə/ | Any mathematical rule expressed symbolically. |
| coordinate verb | /koʊˈɔɹ.dəˌneɪt/ | To place in the same order or rank. |
| calculation noun | /ˌkælkjuˈleɪʃn̩/ | The act or process of calculating. |
Phrasal verbs you will hear
| Word | Pronunciation | Meaning |
|---|---|---|
| call back verb | To request someone's return; to bring back or restore; to recall. | |
| find out verb | To discover, as by asking or investigating. | |
| go back verb | To return to a place or state after having been there at a previous time. | |
| set up verb | /ˌsɛt ˈʌp/ | To make ready for use. |
| step up verb | To increase, augment, or advance, especially by one or more steps. | |
| use up verb | To consume, deplete, expend, or exhaust. | |
| write out verb | To write at full length or in expanded form. |
Sentences worth repeating
Short, complete lines from the video that you can reuse in everyday conversation:
- How does a streaming service know you'll like a movie?
- This answers the question, where can my character go?
- What if we have three moves?
- The question is, what are p's coordinates in our new basis?
- How do you find the new coordinates?
Grammar in this video
The structures the speaker uses most, with the exact words from the video:
| Structure | In the video |
|---|---|
| Present perfect have/has + past participle — a past action that still matters now | You've probably seen · You've seen · You've done |
| Passive voice be + past participle — the focus is on what happens, not who does it | were forced · is called · was born |
| Relative clauses who / which + clause — extra information about a person or thing | squared, which is · that, which is |
Pronunciation to watch
The speaker uses 52 contractions and reduced forms, such as we're, you've, isn't. Say them the short way, as you hear them.
- The “th” sounds: mathematics /mæθ(.ə)ˈmæt.ɪks/, algorithm /ˈælɡəɹɪðm̩/, mathematical /ˌmæθ(.ə)ˈmæt.ɪ.kəl/, theta /ˈθiːtə/
- The “sh” and “zh” sounds: rotation /ɹoʊˈteɪʃən/, transformation /ˌtɹæns.fɔɹˈmeɪ.ʃən/, calculation /ˌkælkjuˈleɪʃn̩/, equation /ɪˈkweɪ.ʒən/, efficient /ɪˈfɪʃənt/
- Long words — get the stress right: transformation /ˌtɹæns.fɔɹˈmeɪ.ʃən/, coordinate /koʊˈɔɹ.dəˌneɪt/, mathematics /mæθ(.ə)ˈmæt.ɪks/, intuition /ˌɪn.tjuːˈɪʃ.ən/, analogy /əˈnæləd͡ʒi/
How to practise with this video
- Listen to the whole video once without speaking and note the words you do not know.
- Shadow it sentence by sentence at normal speed, repeating each one until your rhythm matches the speaker.
- Record yourself and compare with the original, paying attention to words like vector, matrix, minus.
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.