Orthogonal matrices preserve angles and lengths | Linear Algebra | Khan Academy

Showing that orthogonal matrices preserve angles and lengths

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[renesax]

Does sal (Khan Academy) have any videos on SVD?

[bewarethebeef]

Kind of weird that you switched the order of v and w when you wanted to represent the dot product as a multiplication of the transpose of one with the other, right? Or am I missing something?

[kmsakthivel]

Excellent Explaination...purpose served... Thank you

[MrMikael]

Multiplying an orthogonal matrix to a vector can only do 1. nothing 2. flip all of the signs ?

[jasonlin]

Excellent tutorial! Super clear! Thanks!

[geekoist]

I guess on quick way of saying that orthogonal matrices preserve angles and lengths is: orthogonal matrices preserve dot products. Right?

[norwayte]

Could you record a video with all the formulas/outcomes/take-aways and how they relate to each other? A commented overview so to speak for understanding the coherence and for the ability to use it for applications. Thank you.

[DrAituar]

Could somebody please advise the name of playlist, which contain this video

[zes]

no such thing as interesx or not about it, ts just toolx, no interesx

[Phagocytosis]

I can't be sure off the top of my head, but I don't think that would be sufficient. Necessary, but not sufficient. That is to say, whenever angles and lengths are preserved, dot products would be preserved, but I think you could come up with situations where dot products are preserved but not angles & lengths. This is based on the notion that dot products reflect how much one vector goes in the same direction as another vector; you could get as far at a different angle and a different length.