The geometric view on orthogonal projections

Learning Objectives:
1) Given a vector, compute the orthogonal projection onto another vector
This video is part of a Linear Algebra course taught by Dr. Trefor Bazett at the University of Cincinnati

[@vocal.unrest]

This blew my mind and explained it way better than the textbook. Thank you!

[@z.m.4331]

Thank you for the clear explanation, you are the best "textbook companion" ❤

[@DrTrefor]

You're very welcome!

[@mostafaahmadi4998]

Just beautiful and elegant. Incredible explanation. Thank you so much.

[@duncanweakley7525]

Love your enthusiasm as you explain, thanks for your content :)

[@lukewood5751]

Explained better than I've seen it elsewhere. I'd be interested to see you explain it in terms of inner product spaces as well. thanks for the great video

[@anthonyalvarez9272]

Best explanation seen so far. Actually the goat

[@hyperduality2838]

Affine projective geometry is dual to hyperbolic projective geometry. Infinite (affine, canonical or standard) is dual to finite (hyperbolic or non standard). Duality creates reality!

[@SalehGoodarzian]

Thank you very much. Such a great video, very helpful and educational. Best wishes to you

[@muhammadarslanbhatti2139]

beautiful

[@manishajain6038]

Hello Sir, I have one doubt why do we want that y = y^ + z? Is it coming from Pythagoras theorem, but that has to be square if we are using Pythagoras theorem? Am I missing some result that you are using here?

[@luisbelgois2102]

Because is a decomposition of y. y^ is the component of y in the u direction and z is the ortogonal component of y respect to u

[@beetlesstrengthandpower1890]

At the end x2 and x3 are given as vectors in the formula, but they are scalars right?

[@youssefdirani]

You're right

[@mortezakhoshbin]

Thanks alot.are the unit vector of spherical or cylendrical coordinates orthognal?

[@Justkiddingbutuknow]

I've a question can u plz emplain to me why do we need to study orthogonal projection I mean what can we do with it??

[@lucrece4836]

dont know if youre still asking but its a very crucial topic. Often we get a system of equation Ax=b, that cant be solved, since it overdetermined. this means, that b isn't in the column space of A. But as this occurs often, we can't be satified with no solution. So what we do is a projection of b on the columnspace and then solve x, thus obtaining not the exact but best possible solution available

[@Darisiabgal7573]

You always get such glowing compliments but I finally had to give up on your series. 1. Sometimes you talk incredibly fast, and you run your syllables together, i have to go back and rerun some sections 2 or three times over. 2. You cannot see skyblue markers on a white back ground. 3. You write so small that even on an IPAD a alpha looks like a plus, you hats look like your vector marks and in some cases the characters are so small I have to blow up the image. What that means if I am wasting alot of time trying to figure out what you are writing I'm not really understanding what you are saying particularly when you are rambling off stuff 100 MPH.