Advanced Linear Algebra, Lecture 1.4: Quotient spaces

Advanced Linear Algebra, Lecture 1.4: Quotient spaces

If two vectors x and z differ by an element y in a subspace Y, then we say that x≡z (mod Y). This defines an equivalence relation, and the equivalence classes form a vector space called the quotient space of X modulo Y, and denoted X/Y. We define addition and scalar multiplication in this space and show that it is well-defined, as well as discuss what that means. We give some examples of quotient spaces, and prove several basic theorems, such as dim(Y)+dim(X/Y)=dim(Z), and dim(U+V)=dim(U)+dim(V)-dim(U⋂V).

[A.Shafei]

@8:47
"So now if you don't have a good picture as to why we care about this, it might just look like a bunch of notation."

And you proceed to clarify what this concept *is actually saying*

Goddamn! This is so good! I'm risking sounding weird and emotional right now, but this made me feel so good!

[CoDfanatic]

Fantastic as with the group theory playlist... Hope to see a presentation on Jordan Canonical Form one day

[qwing]

fyi the course webpage link does not work

[paulashurov]

For Theorem 1.6, could we also prove this by showing that Y and X mod Y are compliments of each other? Is that even true?