Linear transformations | Matrix transformations | Linear Algebra | Khan Academy
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- Опубліковано 7 лют 2025
- Introduction to linear transformations
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Linear Algebra on Khan Academy: Have you ever wondered what the difference is between speed and velocity? Ever try to visualize in four dimensions or six or seven? Linear algebra describes things in two dimensions, but many of the concepts can be extended into three, four or more. Linear algebra implies two dimensional reasoning, however, the concepts covered in linear algebra provide the basis for multi-dimensional representations of mathematical reasoning. Matrices, vectors, vector spaces, transformations, eigenvectors/values all help us to visualize and understand multi dimensional concepts. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq) so don't confuse this with regular high school algebra.
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4 people linearly transformed them selves from college students to college dropouts
add me to that list
@@Mastblood dont drop out...
@@Mastblood the world needs you
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😆 🤣 😂 😹 😆 🤣 LOL!@
Its awesome how you break down everything in Linear Algebra into understandable modules. When my lecturer teaches, everything is like encoded in alien codes but wen i come here.... i kinda understand it all. Big up Sir. You are my tutor.
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of all math courses at university linear algebra involves the most memorizing and least understanding of why formulas work the way they do. I much prefer calculus and stats where formulas make more sense and thanks for helping out with linear algebra your videos are good
I agree on that,stats and calculus are way better,they make more sense
+Marcus Fenix As a very visual learner, i completely agree with you. They have next to no images or visualizations to give you a better understanding of why it works and how it relates to the actual vector spaces themselves
True AF. Like it will eventually comes down to solving a linear system or matrix, why bother having all of the bullshits like basis, coordinates, linear transformation,...
@dragonzito mojado Well that is kinda why a lot of people are there tho, to understand it
These videos are better than a lot of Linear Algebra courses at universities.
I could just watch MIT Open Courseware and you all day, buy all the textbooks and I wouldn't need university. I could be educated as an engineer from UA-cam and readings alone. That's remarkable.
Maybe for the math and science but you're better off having someone supervising you when learning how to opperate the machinery, also a big part of engineering is team work which is what the labs are good for. Online tutorials are a great suppliment for learning and I use them all the time, but I wouldn't regard them as a substitute. University also teaches you to meet deadlines and work under pressure, something one may not be so inclined to do when studying independently :P
You have a point. Thanks for assuring my confidence in my Ryerson education :D
Mix that in with a tutor or a person you can interact with in person and you might be pretty good off, however I think one of the lost points of a university is that people you can bounce ideas off you and therefore create new ideas. Maybe this doesn't happen as much in undergrad but I think some people don't realize that is what the original purpose of a University is for.
what about labs
what a time to be alive huh lol
After hours and hours of reading my textbook and looking at other youtube videos, this has been BY FAR the best explanation of/how to apply a linear transformation!
Thanks for doing what you're doing Sal! :) This video was posted 9 years ago and it's awesome seeing how many people you've helped. You are going to save my grades for a midterm tomorrow haha.
You speak a language anyone can understand, this is just perfect!
Being here after 14 years is wild 😂🎉❤
Your videos are a great help in studying for my upcoming exam. Thank you very much for explaining the concepts so clearly.
"What is our... Let me do a color I haven't used in a long timeWHITE!" 8:19
i have been strugling with this concept since my college lecture, but this man cleared it all... very well explained
I'm also obsessed with linear independence.
Learned more from Sals linear algebra videos than I did in college
iv been studying for an exam, this has helped loads!
THANKS sir I can understand 😀👍🏻
khanacadey I'd like to thank you. I had a lot of trouble with linear transformations that I couldn't figure it out with your guidance. I truly appreciate it.
Thank you very much for taking your time to break this down
Thank you sooo much, you just made this a lot easier!
the explanation is so clear that i could understood it in english (my native language is spanish).
good job.
I can't believe how well you explain these topics if I can understand this anyone should be able to because I'm not that bright I'm pretty dumb compared to the average but it all clicks so well now.
Big ups to da man making these bad boys, bloke is absolutely carrying me rn
Amazing work! Thanks!
20 minutes here worth more then any number of hours in a lecture hall.
Thank god you exist...
You are better than my University Lecturer , #I got the concept within 13 minute , thanks a lot Sir
thank you for everything mr. khan
Sal, you are a life saver. Thank you
whenever i find u about any topic in youtube then everything gonna seem like simple
Thanks a million
God Bless You
@Khan Academy - So the example to demonstrate where the linear transformation is not valid is one where we did a non linear operation on the variables. So this way, i should assume ...L.T. to be valid in most cases where there is an absence of any explicit non-linear operation on the variables. ?
sir, your the best teacher...ever,,,
whenever I hear this voice I know I am gonna learn something useful
It is very nice lesson .Thank you sir.
Thanks, this has been most helpful!
Amazing explaination
After mapping at 2:55 minutes you put (x1+x2, 3x1) is 3x1 an arbitrary value you have taken or it has a significance
What is the reason behind linear transformation? Why are we transforming these vectors? I get the calculations but I think it's useless if I don't know why I'm doing this
Maybe it is useful in quantum mechanics. Like in Qubit (transforming state |0> to state |1> and vice versa) or in Quantum Teleportation where unitary operator can be useful for transformation. Vectors in QM is considered as state.
thank you so much❤
a nice and clear video. thank you.
I finished my engineering graduation without ever understanding linear transformation because I don't get its meaning in real life. I am sorry but this did not help either. I know how to do the math, I just don't know what it means.
if you not find way for implement this math maybe you will like to look about machine learning and other ai development that require math
It's the matrices, man. A linear application is a matrix.
These were basically just proofs into how to determine if a transformation is linear or not. The applications and more detailed in the next video so check that out and maybe it'll become more clear
Try 3b1b. That's a goldmine in UA-cam.
Moreover, why should anything necessarily have a meaning or purpose to it?
I was having hard time to prove whether the system is linear transformation or not but after seeing your videoes I made my concept clear because of your fantastic explanation I don't know how many tims I should say thank you so much
definitely its a very good work...I appreciated the work since it makes me very clearly notice a concept of linear transformation....you r now be my tutor..
A video can save you from failing the exam. Thanks to khan academy!
Anyone from 2024 watching this?
Yes
The person teaching is probably a black man also na the stupid course way I get 😂
Ya boy
😂😂
8 hours away from failing lineal algebra 😭
you sir are a gentlemen, and scholar.
THx alot!!! you save my life!!!
I was at matrix section for high school preparation and there comes transformation to which I don't understand some and I started searching and then boom this, it seriously got me doubting 'do I really know all this' only after coming to UA-cam and reading the down message did I get it's for majors. I think it's important to mention the prerequisite in the main website.
Could you comment on the difference between a transformation from R2 to R2 and R to R. I am referring to your comments around 10:17....
It means keeping transformation of vectors locked into 2D plane, it can't go higher or lower than 2D.
Generalized it means RN to RN transformation is N-D vectors can't go any higher or lower than N-Dimension
thank you
thank u soo much sal!
c can be a complex number as well, and the components of vectors a and b can also be complex.
thank you so much man!! this was so helpful :)
Q4: Determine whether 𝑇: 𝑅2 → 𝑅2 is a linear operator.
𝑇(𝑥, 𝑦) = (𝑥, 0)
Sir, u r just so helpful... 😁😁thanks for being there for students like me...
It sounds straightforward; however, it is still missing to answer why we transform? What is the purpose of transformation?
Dear Khanacdemy. First of all great video's and perfect explanation.
I have a little question here: doesn't the first condition imply the second?
if T(a+b) = T(a) + T(b) => T(a+a+....a) = T(a) + T(a) + ... + T(a) = c.T(a) ?
domien van steendam Depending on the Universe we choose it implies it, yes. It works quite well on the integers, but you couldn't make that implication when dealing with rational values for c, for example.
Bro I am taking this in middle school insane
great!
thank you for share
two minutes in you already explained this better than my professor.
He is usually very good but I was pretty lost on what exactly was going on here.
ITS OVER 9000!
Wow ❤
THANK YOU
I’m proud of myself bc I’m a freshman doing this
You are AWESOME dude!
Thank you!!!!!
thanks
Gracias!
1.3 mln views and just 5.6k likes? People are so ungrateful. They watch these amazing tutorials for free and don`t even press the like button((
Wow! an amazing lecture. quite interesting pal!
Thank you Sal
are you alive
Could you please let me know when the term "Linear Fractional Transformation" was used for the first time (year, decade, century)? Thank you!
very nice oh yes
ty
God bless you!!!
Does Khan Academy provide assignments?
gracias academia khan por los subtitulos asi aprendo mucho =D
Please,suggest me about differences between operators and functions.
Yes
This guy is totally the Bob Ross of linear algebra.
could i get a recommended order of how to view all of your videos on linear transformations? anyone? thanks.
Excelent explanation, simple and efective. How did you do it?
What software did you use to capture and write?
Sorry my poor english. Regard.
Anyone from 2025 watching this video?
me
Yep
Lol
very good
good
how to find the liner transform vector it self before checking ?
Anyone from 2020?
i am from 2024
felt like i fully understood it after this video, where as class lectures made me feel lost. Thanks!!!!
You need to watch the 2 subsequent videos as well since they build up the concept.
2:05 - 2:37 i cant stop laughing at this LMAO, it reminds me of XQC's text to speech HAHAHAH
looool
LLLLLllLlLLlLLlLllLLlLLLllllLllL
Is this for Algebra 1
What software do you use to make your videos?
You deserve to go to heaven
what would y=2x+5 be in linear transformation?like horizontal stretch and that kind of stuff
i am confuse what does X1 and X2 represent? are they the scalar?
Before that I was saying, ALGEBRA is too easy 😢
4:40 brain is malfunctioning ... 😁😁
If A is a 2x2 matrix such that tr(A)=3 and tr(A^2)=5 , SO HOW TO FIND det(A)
2
And the previous too???
Thank you so much for making this video! I came here really confused in this topic and you made it real clear with your examples!
Only thing I could advice you about is that in the first example I got kinda confused 'cause of the way you wrote down the exercise and you looked back at the reference of the two rules, at some point it looked a little messy, maybe you could organize a little more your notes. The colours helped though, again, good job :3
At 12:23 how come (ca1)^2 isn't C^2+2CA1+A1^2?
Shouldn't 3 x1 = 3a1 + 3a2? Why did he wrote 3x1 = 3a1+3b1?