Vectors Spaces - The Definition - 3 problems

In this video I just run through the definition of a vector space. At the end of the video there are 3 sets for which we will decide whether or not produce vectors spaces depending on how we define addition and multiplication. I encourage you to try to answer the questions on your own and check my work in the following videos.

[vanguard]

More linear algebra please :) especially things like nullity, rank, basis, eigenvalues, eigenvectors, etc.

There's literally no good videos on youtube covering these abstract linear algebra concepts :(

[darkdingo]

3rd year watching you solve math problems +patrickJMT . Calculus, HL math, and now Lin. Alg. You're the bestttt!

[mgchevelle]

The most solid math help videos on the internet! You have helped me a lot back when I took calculus and linear algebra and I still find myself coming back to your videos for quick reviews as I enter my last semester as an undergrad physics major. Thank you!

[joedoceihill]

Linear algebra!! Thank you! More videos please!

[boody]

I got A+ in liner algebra course, thanks to you ...

[christianhitrancis]

i was lost in class during this... thanks for explaining it bro

[AliRaja]

linear algebra 1 sucks.
Thanks Patrick for the rescue.

[luiss.]

You are a beast in linear why didn't i find your videos earlier in the semester ):

[KingofHearts]

I don't understand this "not normal addition" business, how do you define *x*+ *y* to = *xy*? Would *x* and *y* be vectors only containing quantities such that the sum and product of those quantities are equal? The only number I can think of that fulfills that is 2, so would x and y just be vectors filled with 2? Also, how do I test that the axioms hold true for *x*+*y* = xy? for A1 would I say that x+y = y+x = xy = yx? in which case that would hold true right? I have a problem that says that [x1, x2]^T + [y1, y2]^T = [0, x2+y2 ]^T, how do I start testing the axioms? If I were to test the first one is this the right thought process:

*x* + *y* = *y* + *x*

so

[y1, y2]^T + [x1, x2]^T = [0, y2+x2 ]^T = [0, x2+y2 ]^T

So then A1 holds true?

[landonazbill]

Will you ever consider doing topology or group theory?

[diwakarkumar]

Is there any playlist for this Linear Algebra thing covering Matrices, Vector spaces, its dependencies, eigen values etc ?

[banditto]

any chance on posting the 3rd problem? that one doesn't make sense to me.

[EccoMath]

I see Vector Space Yes or No Example 1 and 2, but where is Example 3? Does anyone have the link for that?

[bremulate]

What about the vector inverse. vec.u x vec.u^-1 = 1. That does not apply?

[khoinguyen]

I hated vector space you have to look at so many axioms and my teacher didn't care if one of them failed he wanted us to check ALL lmao but it only makes sense for us to understand how every single axiom works

[ajkan]

i thought that plus and multiplication sign were xor and nxor gates hahaha

[-lv]

What does he mean when he talks about all the types in a vector are x x or x y?

[thebossful]

cant find the answer to the 3rd problem

[MuffinsAPlenty]

It is interesting to note that the commutativity of addition is unnecessary as an axiom. It can be proven from the other axioms.

Let x and y be any two vectors.

Then consider (1+1)(x+y).

By axiom 7, (1+1)(x+y) = (1+1)x + (1+1)y, and by axioms 8 and 10, (1+1)x + (1+1)y = (x + x) + (y + y) = x + x + y + y by axiom 3.
Now, by axioms 8 and 10, (1+1)(x+y) = (x+y) + (x+y) = x + y + x + y by axiom 3.

So we see that x + x + y + y = x + y + x + y

By axiom 5, we get −x + x + x + y + y = −x + x + y + x + y
Also by axiom 5 and 3, we get 0 + x + y + y = 0 + y + x + y
By axiom 4 we get x + y + y = y + x + y

By axiom 5, we get x + y + y + (−y) = y + x + y + (−y)
Also by axiom 5 and 3, we get x + y + 0 = y + x + 0
By axiom 4, we get x + y = y + x

And technically, we should state that axioms 1 and 6 are technically used in this proof too by virtue of adding and multiplying vectors together.

So we can prove axiom 2 (the commutatvity of addition) by using axioms 1, 3, 4, 5, 6, 7, 8, and 10.

That being said, I think it is helpful for pedagogical reasons to include it as an axiom of a vector space, even though it is technically unnecessary.