Orthogonal Decomposition Theorem Part II

Description: Given a vector and a subspace, we decompose the vector (uniquely!) as a the sum of a vector in the subspace and a vector in the orthogonal compliment to the subspace. We prove this theorem and show the geometric intuition.

Learning Objective:
1) Give the orthogonal decomposition of a vector

This video is part of a Linear Algebra course taught by Dr. Trefor Bazett at the University of Cincinnati

[AshwinSajeev]

I love this video. You have literally changed my life

[AhmedHanafyAbulwafaBayumy]

Hello!
is there a complementary video to this one? it just ended all at once without completing the last note.

[weisanpang]

Is this a norm ? "n" comes before "p" in alphabetical order, but in the description of the subspace decomposition, u1...up are the basis vectors that live in the smaller subspace W, while the larger subspace lives in "n", as in the basis vectors would be (un comes after up ?) ? Why not make it the other way round ?

[confused]

I think you should have started with a diagram before you go into the mathematical theorems. Thank you!
It is - for me - to follow the theorems that you described without seeing what they are,

[hasanmohamadi]

thank you very much you are a fantastic teacher!

[mostafaahmadi]

Writing a description of what the video will be about and more importantly the learning objectives in the description box are very helpful. You could be watching a video packed with a lot of mathematical notations and manipulations that you sometimes lose the track of the intuition and the goal behind the video. But going back to the description and learning objectives AFTER you have watched a video will bring all the ideas of the video together and everything suddenly clicks and you become so happy. Thank you very much for including the description and learning objectives for every video in this playlist. Please kindly keep doing it for your future videos. Thank you

[venumadhavrallapalli]

Doubt: If the y hat vector belongs to a subscpace W with p-dimensions, and z vectors belongs to subspace Wperp, then what is the dimesnion of this Wperp subspace. I saw in textbook that dimension of Wperp is n-p, n being the dimension of the bigger vector space. But how is it n-p, I dint understand, asfor my understanding even Wperp should be of same dimension as W subspace, Please clarify this !!

[priyaldesai]

Thank you so much for the proof Professor!

[Michellefilipaa]

Is it just me or did this video kinda end abruptly ... I didn't get the note :((