I thought I knew linear algebra very well from my courses at university, but you have given me a whole new geometric perspective. I love the way you motivate these concepts. Your criticism of the nomenclature ‘orthogonal matrix’, ‘symmetric matrix’ is original and illuminating. Thank you for taking the time to make these videos.
I'm a little confused as to why you keep on saying that the Orthoscaling (aka symmetric) transformation is a stretch. The way I see it, it's only a stretch of the eigenvectors of the symmetric matrix. All other vectors get knocked off their span. So, isn't an orthoscaling transformation technically a rotation as well? I mean, if we multiply a vector by an arbitrary symmetric matrix, that vector will probably be rotated AND scaled...
@@MathTheBeautiful okay! It took me a while to understand what u meant by 2 simultanious, but now I understand what u mean - one scalikg action for each of the orthogonal eigenvectors of the new basis!
the only way i view of Matrix being orthogonal is , the three orthogonal vectors after transformation still holds their orthogonaility, until just learned that symmetric does that too.
i believe that the name orthogonal transformation comes from the fact that a vector that does not change it's lenght varies "orthogonally" since it's derivative is orthogonal to the instantaneous vector.
I don't know if you have time, but if you do, I would really, really appreciate an answer from you to this question I posted on Reddit: I feel like I'm close to understanding it geometrically, but I'm still confused. Most answers are using too much terminology - I'm still having trouble with symmetric transformations intuitively (the whole circle to ellipse thing of linear transformations is confusing me actually). Here, I'll post it below: www.reddit.com/r/3Blue1Brown/comments/banhrq/linear_algebra_question/ Thank you!
Hi Joshua, I hope that the answer that I'll give you doesn't upset you, for it might make you feel like you have a lot of work in front of you. And it's true, but it's very fun and fulfilling work. I know exactly what you're asking and it's a great question. And I know that you're deeply dissatisfied with the answers you received on reddit. Isn't it amazing how an exhaustive and technically correct answer can be completely unsatisfying. So I'll get you started by inviting you to clarify your question to yourself. What is the space? What is the transformation? What is the inner product? What is the meaning of the transpose symbol? Is the identity you present a statement that's always true or a statement about a special property of A? Or is it the definition of the transpose symbol? Or is it the property of the inner product? Or do you not have an inner product and just asking a question about matrix multiplication? Note that each one of these questions can be answered in a number of ways. In math, and LA in particular, the expressions that we write on a piece of paper can mean wildly different things depending on the interpretation. By the time you clarify your own question to yourself, you won't need anybody's help answering it. Pavel
@@MathTheBeautiful Pavel - thank you so much for the long response - it was the opposite of upsetting. You're right - technically correct answers have been dissatisfying, but its probably mostly my fault for not truly understanding what the "technical terms" really mean geometrically. I'm gonna tackle each part of your answer - I'll get back to you soon! Thank you again! - Josh
Go to LEM.MA/LA for videos, exercises, and to ask us questions directly.
I thought I knew linear algebra very well from my courses at university,
but you have given me a whole new geometric perspective.
I love the way you motivate these concepts. Your criticism of the nomenclature ‘orthogonal matrix’, ‘symmetric matrix’ is original and illuminating.
Thank you for taking the time to make these videos.
Thank you for letting me know. Much appreciated.
Thanks to you, 3Blue1Brown, and Strang, Linear Algebra has become one of the most beautiful topics I've ever studied. Thank you!
I'm a little confused as to why you keep on saying that the Orthoscaling (aka symmetric) transformation is a stretch. The way I see it, it's only a stretch of the eigenvectors of the symmetric matrix. All other vectors get knocked off their span. So, isn't an orthoscaling transformation technically a rotation as well? I mean, if we multiply a vector by an arbitrary symmetric matrix, that vector will probably be rotated AND scaled...
Yes, you are exactly right. Two simultaneous orthogonal stretches combine to a distortion.
@@MathTheBeautiful okay! It took me a while to understand what u meant by 2 simultanious, but now I understand what u mean - one scalikg action for each of the orthogonal eigenvectors of the new basis!
What a beautiful explanation.
Hahaha, I have to say you are brilliant :D Nobody has inspired me more about curvature than your propulsion of chalk! I have been hooked. Well Done :)
you are a blessing in my life
Thank you, that means a lot.
@@MathTheBeautiful I have started loving this subjects because of you sir.A very big thankyou sir.
Great explanation, thanks!
the only way i view of Matrix being orthogonal is , the three orthogonal vectors after transformation still holds their orthogonaility, until just learned that symmetric does that too.
i believe that the name orthogonal transformation comes from the fact that a vector that does not change it's lenght varies "orthogonally" since it's derivative is orthogonal to the instantaneous vector.
Hello, do you cover gram schmidt ortogonalisation? Btw love your channel!
Not yet, but coming soon.
I don't know if you have time, but if you do, I would really, really appreciate an answer from you to this question I posted on Reddit: I feel like I'm close to understanding it geometrically, but I'm still confused. Most answers are using too much terminology - I'm still having trouble with symmetric transformations intuitively (the whole circle to ellipse thing of linear transformations is confusing me actually). Here, I'll post it below:
www.reddit.com/r/3Blue1Brown/comments/banhrq/linear_algebra_question/
Thank you!
Hi Joshua,
I hope that the answer that I'll give you doesn't upset you, for it might make you feel like you have a lot of work in front of you. And it's true, but it's very fun and fulfilling work. I know exactly what you're asking and it's a great question. And I know that you're deeply dissatisfied with the answers you received on reddit. Isn't it amazing how an exhaustive and technically correct answer can be completely unsatisfying.
So I'll get you started by inviting you to clarify your question to yourself. What is the space? What is the transformation? What is the inner product? What is the meaning of the transpose symbol? Is the identity you present a statement that's always true or a statement about a special property of A? Or is it the definition of the transpose symbol? Or is it the property of the inner product? Or do you not have an inner product and just asking a question about matrix multiplication? Note that each one of these questions can be answered in a number of ways. In math, and LA in particular, the expressions that we write on a piece of paper can mean wildly different things depending on the interpretation. By the time you clarify your own question to yourself, you won't need anybody's help answering it.
Pavel
@@MathTheBeautiful Pavel - thank you so much for the long response - it was the opposite of upsetting. You're right - technically correct answers have been dissatisfying, but its probably mostly my fault for not truly understanding what the "technical terms" really mean geometrically. I'm gonna tackle each part of your answer - I'll get back to you soon!
Thank you again!
- Josh
Hi Josh, it's not you, it's the author of what you're reading. It to me a long time to realize that myself and it was very liberating when I did.