Subspace Help: Properties & Verifying Examples

[MidgetDwarf]

Unfortunately, this is the only book I can afford for a while. Though it might not be an optimal one, I figured that with some help, I could work through it and get a better book down the line.
I'm counting on it!
So:
Since, by the (1)additive identity, 0 is an element of the set
and, since the subspace must be (2)closed under addition (u, v, (u+v)EU),
and, since the subspace must be (3)closed under scalar multiplication (auEU if aEF),

then:
Couldn't any of the other x values = b so that (2) holds true?

Ie. x_2 = b

Thanks for your patience, everyone. I know I have a weak grasp and I want to move forward with strong footing, and you're all invaluable to my learning in LA...

Linear Algebra Done Right is a very clear and concise book. It is a good book. But sometimes, people may not have the prerequisites or experience to enjoy such books. There is nothing wrong with reading an easier book, then coming back to another.

The only requirements for Axler, is being able to read and write basic proofs. Have you taken a course in logic? Are you familiar with how to read and write proofs?

 

[glauss]

@glauss Here's an example proof from another thread, to let you see what I mean by mathematical logic and proofs:

Prove that the equation $x^2+(2m+1)x+(2n+1) = 0$ does not possesses any rational roots if $m \in \mathbb{Z}, n \in \mathbb{Z}$.

Outline Proof (with exercises left to the reader!):

1) If the equation $x^2 + (2m + 1)x + 2n + 1 = 0$ has a rational solution, then that solution must be an integer. (Exercise for you.)

2) Let $r \in \mathbb Z$ be a solution. And, we have: $r^2 + (2m + 1)r + 2n +1 = 0$ and, rearranging we have $r(r + 2m + 1) = -(2n +1)$

3) Note that $-(2n+1)$ is odd.

4) If $r$ is odd then ... contradiction. (Exercise)

5) If $r$ is even ... contradiction. (Exercise)

6) The equation cannot have an integer solution, therefore cannot have a rational solution, therefore any solution is irrational.

QED

If you can follow that, then fine; otherwise, you've a lot of work to do before you begin Axler.

I will give that linked PDF a read, on matricies and group theory, it only looks like a few pages.
Do you have another recommendation for logic and proofs? I’d like to bring my skills up to par quickly and get back to working through Axler ASAP...

 

[fresh_42]

We have a forum
https://www.physicsforums.com/forums/math-proof-training-and-practice.296/
where you can find hundreds of examples and where you can post questions to specific proofs or methods in general.

 

[PeroK]

I will give that linked PDF a read, on matricies and group theory, it only looks like a few pages.
Do you have another recommendation for logic and proofs? I’d like to bring my skills up to par quickly and get back to working through Axler ASAP...

Logic and mathematical proofs can be difficult to learn. I have two elementary undergraduate maths books, one on abstract algebra and on real analysis, which cover the material to some extent.

Often, for physics students, Linear Algebra is their first exposure to formal proofs. That's why the right textbook is important. You could try this one. It's certainty very different from Axler.

http://www.math.byu.edu/~klkuttle/0000ElemLinearalgebratoprint.pdf

There are lots of proofs in there. In fact, I think he proves almost everything!

Ultimately, it's just practice, practice, practice. That said, in my view, maths is a bit like music: some people understand it naturally and intuitively; and some people have to work at it very hard indeed!

 

[cgrie]

@glauss Here's an example proof from another thread, to let you see what I mean by mathematical logic and proofs:

Prove that the equation $x^2+(2m+1)x+(2n+1) = 0$ does not possesses any rational roots if $m \in \mathbb{Z}, n \in \mathbb{Z}$.

Outline Proof (with exercises left to the reader!):

1) If the equation $x^2 + (2m + 1)x + 2n + 1 = 0$ has a rational solution, then that solution must be an integer. (Exercise for you.)

2) Let $r \in \mathbb Z$ be a solution. And, we have: $r^2 + (2m + 1)r + 2n +1 = 0$ and, rearranging we have $r(r + 2m + 1) = -(2n +1)$

3) Note that $-(2n+1)$ is odd.

4) If $r$ is odd then ... contradiction. (Exercise)

5) If $r$ is even ... contradiction. (Exercise)

6) The equation cannot have an integer solution, therefore cannot have a rational solution, therefore any solution is irrational.

QED

If you can follow that, then fine; otherwise, you've a lot of work to do before you begin Axler.

I took linear algebra without proofs before I took an intro to proofs class. In my opinion, there were lots and lots of definitions to learn in even introductory linear algebra, and the extra terminology can get frustrating. It's really nice to start out with the easiest examples until you get the hang of how to use contradiction, direct, etc.

 

[MidgetDwarf]

Hmm there's also Anton. Its more applied, but proofs are gi

I will give that linked PDF a read, on matricies and group theory, it only looks like a few pages.
Do you have another recommendation for logic and proofs? I’d like to bring my skills up to par quickly and get back to working through Axler ASAP...

https://www.people.vcu.edu/~rhammack/BookOfProof/

Here you go.. I found this book much clearer than the Velheim book. Solutions are given to some problems in the back. I am not familiar with this edition, but it should still be the same.