Linear Algebra 27 | Dimension of a Subspace [dark version]

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This is my video series about Linear Algebra. I hope that it will help everyone who wants to learn about it.

#LinearAlgebra
#Vectors
#Matrices
#MachineLearning
#Eigenvalues
#Calculus
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(This explanation fits to lectures for students in their first year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[starinsky]

You are doing great work for math students at university who skip class :) I must support your channel in future. Also thanks for doing such a great work.

[wmansour]

Hey! Thank you professor for the amazing content,

I have a question though, about the Exchange Lemma. does the set V have to be a Basis? My understanding is and please correct me, if the set V is a basis, we can only exchange but not add any vectors as any new vectors added will make the basis set linearly dependent. I think V should in this situation be just a set spanning the subspace U not a basis per se.


Edit: nvm I think i got it, we are adding *from* the basis set to the Lin Independent set to *create* a new basis, but then the new set will not be a basis unless we remove all the vectors from that linearly independent set that were rendered dependent by the new basis? my question here is that if we exchange a vector that makes more than one vectors in the new set dependent will the one-by-one exchange work in the end?