linear algebra vector space (25 examples)

Vector Spaces. Definition and 25 examples. Featuring Span and Nul. Hopefully after this video vector spaces won't seem so mysterious any more!

[nathanisbored]

Thank you! I failed linear algebra last time I took it and I think when we started talking about vector spaces was when everything started to go way over my head, but now I understand it a lot better!

[neptunian]

I think one of the coolest things is that you can always equip the set of all homomorphisms from one vector space to another with an addition and scalar multiplication such that they form a vector space.

[CornishMiner]

Very useful video. Students often get bogged down in the checking and the proofs, yet this insight is so important.

[purple]

My cousin sent me this saying you looked like the male version of me, at first I was upset as recently someone said I looked like a brown snape, but after skimming your video your mannerisms made me feel nice and I'm sure make vector spaces more digestible. Thanks for the good content twin!!

[fatimatulzahra]

U are the best teacher ever I saw. I love your mathod. Sir keep it up. Love from Pakistan

[tacticalmistress]

Nice simplicity you put to the subject. Thank you for your lectures Dr.

[edwardhuff]

Quantum physics got its vector space mentioned (or close to it, Hilbert space), but General Relativity got left out, as usual. Simple example is the tangent spaces (tangent bundle) to a circle, one vector space (the tangent line) for each point of the circle. For a sphere, the spaces are the tangent planes. Each vector defines a directional derivative for functions of points on the sphere, and the vectors work just like differentials. See Robert Wald's text. And Tensors are defined as linear maps from vectors to reals, etc.

[nynthes]

So if the first three properties hold, all the others in the list automatically hold?

[martinepstein]

For a while the only kind of vectors I knew about were in some sense functions. For instance, a vector in R^3 is a function from {1, 2, 3} to R. An interesting vector space where the vectors are not functions is a quotient space: let V be a vector space and U a subspace. For every v in V we define the "coset"
[v] := {v + u | u in U}
We define the obvious addition and scalar multiplication on cosets:
[v] + [w] := [v + w]
a*[v] := [a*v]
Under these operations the set of cosets is a vector space, denoted V/U. Its vectors are sets of vectors in V.

[MaruBaku]

amazing video clears up all my questions! thank you!

[raulghoora]

I wish you were my lecturer xD. Lectures would never be boring!

[NeonArtzMotionDesigns]

Well u just saved me so I highly appreciate this

[yhmah]

Merci Dr peyam pour ce cours, je vais vous présenter à mes camarades qui ont des problèmes avec l’anglais et l’algèbre avec un grand A :)

Tensor calculus, please :) I would like to learn (as a hobby ...) in that direction, but the topic is not always clear for me (I've tried several youtube lectures on it, maybe I have some idea now, but the Peyam style can boost the learning curve, I bet ....).

[Ensivion]

There is an issue with the vector space of {0, water, syrup, drink} . What is water-drink? They aren't in the set itself, thus it breaks the first/third rule of vector space.

[Vincentsgm]

Mon 1er professeur d'algèbre linéaire nous a introduit les applications linéaires avec un marché de tomates, carottes, et aussi avec des lapins et des chapeaux de magicien, pour le côté abstrait.

[nicolomone]

Dr Peyam, {0, water, syrup, drink} is still a vector space even though syrup+drink=drink ? aren't there two Os ? syrup+drink=drink=0+drink -> syrup=0

[johannesh]

Isn't every field (Körper) also a vector space? e.g. the real numbers.
The ten commandments for vector spaces strongly resemble the axioms of addition and multiplication those must obey

[suhaniahuja]

at 20:20 you say that this is a subspace of a non-vector space (r^2) but isn't r a vector space

[MrRyanroberson]

are functions just a subset of mappings? f(x) = 3 and 2 is not a function because its output isn't one value (perhaps if the set of all values were collected into a vector it could be called a function)