Subspace Help: Properties & Verifying Examples

[glauss]

Summary:: Properties of subspaces and verifying examples

Hi,
My textbook gives some examples relating to subspaces but I am having trouble intuiting them.
Could someone please help me understand the five points they are attempting to convey here (see screenshot).

 

[PeroK]

What don't you understand? Let's start with example a).

 

[glauss]

Thank you for the reply.
Why must b = 0 for example a?
F, in this book, is generalized so that it includes the real and complex fields.

 

[PeroK]

Thank you for the reply.
Why must b = 0 for example a?
F, in this book, is generalized so that it includes the real and complex fields.

To test whether a subset is a subspace, you need to check the axioms. Why don't you do that?

 

[glauss]

Is it simply because b is not an element of F^4?

 

[PeroK]

Is it simply because b is not an element of F^4?

No. $b \in F$

 

[glauss]

But not in F^4 unless it = 0 with the given expression for x3, right? ;)

 

[PeroK]

But not in F^4 unless it = 0 with the given expression for x3, right? ;)

I'm not convinced you understand the set notation employed here. $F^4$ is the set of all 4-tuples of elements in $F$. Think of $\mathbb R^n$. Do you know what that is?

 

[glauss]

Oh I thought F^4 would be four dimensional set of which F is a superset.
I’m actually really dumb so feel free to lead me by the hand to the concepts I’m trying to grasp

 

[PeroK]

Oh I thought F^4 would be four dimensional set of which F is a superset.
I’m actually really dumb so feel free to lead me by the hand to the concepts I’m trying to grasp

Okay, but are you sure you have the mathematical prerequisites to study linear algebra?

 

[glauss]

Yes. Maybe I should delete and re create this thread so I get another person to assist? This is an introductory topic on the course, but I want to understand it well, so I’m asking for an assist

 

[glauss]

I'm not convinced you understand the set notation employed here. $F^4$ is the set of all 4-tuples of elements in $F$. Think of $\mathbb R^n$. Do you know what that is?

Yes, I know what that is

 

[berkeman]

Thread closed temporarily for Moderation...

 

[fresh_42]

Yes, I know what that is

Then please explain what $\mathbf{F}$ is and what $\mathbf{F}^4$ is. It is basically impossible to help you without knowing where you are at to pick you up. @PeroK's question in post #10 was aimed to figure this out. "Yes, I know" is insufficient, esp. because your posts suggested that you do not.

 

[glauss]

I apologize for being difficult.

to be honest, i actually previously considered F to represent F^Infinity, of which F^n would be a subset

 

[fresh_42]

I apologize for being difficult.

to be honest, i actually previously considered F to represent F^Infinity, of which F^n would be a subset

Neither is the case. $\mathbf{F}$ represents the field the scalars are taken from: $b,x_2, etc.$ It is stands e.g. for $\mathbb{R}, \,\mathbb{C},\,\mathbb{Q}$ or any other fields. $\mathbf{F}^4$ is a four dimensional $\mathbf{F}-$linear vector space.

You are right that $\mathbf{F}^\infty$ could be seen as vector space of infinite dimension and $\mathbf{F}^4$ a subspace. However, it would be a) a bad notation, b) meaningless until the embedding is specified, and c) far beyond what the exercise is about.

You cannot add scalars and vectors, they belong to different sets. Exercise (a) is crucial for the understanding of linear spaces. So what makes a subspace a subspace? What must hold? And why doesn't it if $b\neq 0$?

 

[Mark44]

For the problem of part a) you need to show (i.e., prove) two propositions:

1. If b = 0, then $S = \{(x_1, x_2, x_3, x_4) \in \mathbb F^4: x_3 = 5x_4 + b\}$ is a subspace of $\mathbb F^4$.
2. If $S = \{(x_1, x_2, x_3, x_4) \in \mathbb F^4: x_3 = 5x_4 + b\}$ is a subspace of $\mathbb F^4$, then b = 0.

(I gave the set a name: S.
To prove that a set is a subspace of some vector space, all you need to do is to show that addition is closed, and scalar multiplication is closed. IOW, for addition, if u and v are in the set, then u + v is also in the set. For scalar multiplication, if k is a scalar in the field (in F in this problem), and u is in the set, then ku is also in the set.
That's it.

 

[Mayhem]

Page 18, box 1.34 gives you the conditions of a subspace.

For a) what you need to show is that if $b \neq 0$, one or more of these conditions are not satisfied.

Hint: $0$ can refer both to the scalar $0$ and the vector $0$, which for $\mathbb{F^n}$ is $(0,0,...0)$ with $n$ zeroes.

 

[PeroK]

I apologize for being difficult.

to be honest, i actually previously considered F to represent F^Infinity, of which F^n would be a subset

I had a quick look at Axler. He does not include an introductory section on sets and basic mathematical logic and proofs, which is what you may be missing. And, it seems like he jumps into the deep end a little. There are precious few examples of vector spaces in the opening chapter.

This might not be a good book as a first exposure to pure mathematics.

It also looks a touch idiosynchratic to me. Which is what you might expect, given the title.

 

[Mayhem]

This might not be a good book as a first exposure to pure mathematics.

It isn't. There's no way someone new would get through these problems without assistance from the internet.

 

[PeroK]

It isn't. There's no way someone new would get through these problems without assistance from the internet.

Or even with help from the Internet!

 

[Mayhem]

Or even with help from the Internet!

Well, this place offers gratuitous amounts of help. :)

 

[MidgetDwarf]

Thank you for the reply.
Why must b = 0 for example a?
F, in this book, is generalized so that it includes the real and complex fields.

F in the book is denoted as a field. A field has certain properties. Do you know what the properties are? F^n is denoted to be the set of all ordered n-tuples whose ordered entries are those coming from the field.

Ie., R^2 is the set of all elements (a,b) where a , b are in R (the field consisting of real numbers in certain properties that I mentioned if you knew). Moreover, (1,2) is different from (2,1) in R^2 . (why, this goes back to the definition of an n-tuple)

Now, do you understand vector addition? scalar multiplication?

What does it mean for a subset of a vector space to be a subspace?

Depending on what was given as the definition (sometimes called the subspace criterion or something similar),

is the 0 vector of F^4 an element of the set given? Is it closed under addition, scalar multiplication?

 

[Mayhem]

Minor hijack, but as I'm using the same book as OP, he or she might find it useful as well.

If $S$ is a nonempty set and $\mathbb{F}$ is a field, then $\mathbb{F}^S$ means ... ? It has to do with functions and mapping. Can it be translated into the form $f : \mathbb{F} \rightarrow S$ or is it the other way around? For example, $\mathbb{R}^{[0,1]} = f : \mathbb{R} \rightarrow [0,1]$?

 

[glauss]

I had a quick look at Axler. He does not include an introductory section on sets and basic mathematical logic and proofs, which is what you may be missing. And, it seems like he jumps into the deep end a little. There are precious few examples of vector spaces in the opening chapter.

This might not be a good book as a first exposure to pure mathematics.

It also looks a touch idiosynchratic to me. Which is what you might expect, given the title.

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.

Well, this place offers gratuitous amounts of help. :)

I'm counting on it!

For the problem of part a) you need to show (i.e., prove) two propositions:

1. If b = 0, then $S = \{(x_1, x_2, x_3, x_4) \in \mathbb F^4: x_3 = 5x_4 + b\}$ is a subspace of $\mathbb F^4$.
2. If $S = \{(x_1, x_2, x_3, x_4) \in \mathbb F^4: x_3 = 5x_4 + b\}$ is a subspace of $\mathbb F^4$, then b = 0.

(I gave the set a name: S.
To prove that a set is a subspace of some vector space, all you need to do is to show that addition is closed, and scalar multiplication is closed. IOW, for addition, if u and v are in the set, then u + v is also in the set. For scalar multiplication, if k is a scalar in the field (in F in this problem), and u is in the set, then ku is also in the set.
That's 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...

 

[Mayhem]

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 = (1/5)x_4 = 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...

I'll give you a hint. To check if the zero vector ($0 \in U$) is in the subspace, you just plug the zero vector into the conditions of the subspace. Recall, Axler defines $0 = (0,0,...,0)$ where the RHS is a list of length $n$ consisting only of zeros (Definition 1.14, page 7.)

 

[fresh_42]

Unfortunately, this is the only book I can afford for a while.

Alternatives are second hand books, or the modern version: download pdf from the internet. There are hundreds of lecture notes and scripts, even books online, especially for the standard courses.

This does not mean you weren't welcome on PF, you are. Just try to be as specific as you can with what you write, so that we can correct in time what you possibly misunderstood. Unlearning is far more difficult than learning.

And in case you want to write proper code, which would help us a lot, too, see
https://www.physicsforums.com/help/latexhelp/

 

[PeroK]

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...

To do pure maths, you need to develop the technique of logic and proof writing. You're missing a key prerequisite for this book. There are a lots of free pdf's on introduction to abstract algebra. This first one looks quite good:

https://homepages.warwick.ac.uk/~maseap/teaching/aa/aanotes.pdf

Chapter III is an intro to matrices and linear algebra, before he moves on to group theory.

You could also have a look around for free pdf's on introduction to linear algebra. I found this one, which looks a bit more introductory than Axler:

https://www.math.ucdavis.edu/~linear/linear-guest.pdf

That said, if you have no experience of pure mathematics, then it's tough. I'd look for an alternative, as I think Axler is not the right book for you. Not yet anyway.

 

[Mark44]

So:
Since, by the (1)additive identity, 0 is an element of the set

I assume you're working on item 1 of the two statements I listed; i.e., starting with the assumption that b = 0. If so, yes, it's clear that (0, 0, 0, 0) is an element of the set that I called S in this case.

and, since the subspace must be (2)closed under addition (u, v, (u+v)EU),

What does this notation mean -- (u, v, (u+v)EU) ?
To prove this part, still assuming that b = 0, you need to show that if u and v are arbitrary vectors in S, then $u + v \in S$.

and, since the subspace must be (3)closed under scalar multiplication (auEU if aEF),

I don't know what this notation means, either -- auEU if aEF.
Still assuming that b = 0, what you need to show is that if k is an arbitrary element of the field F, and u is an arbitrary element of S, then $ku \in S$.

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

No, because the statements must hold for any arbitrary vectors in S.

What I've laid out above is the proof of one direction in the if and only if proposition (i.e., assuming b = 0). Next you have to proved the converse. IOW, if S is a subspace, then b = 0.

 

[PeroK]

@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.