Vector spaces

[jsbeckton]

Here is the 2 part question:
Consider the revector space C[1,0]. In each case, show that the subset W either is or isn't a subspace.

a) Let W be the subset of C[0,1] consisting of all functions f such that f(0) = 1
I believe that this is not a subspace becase it does not satisfy the 0 vector

b) Let W be the subset of C[0,1] consisting of all functions f such that the (intergral from 0 to 1) f(x) dx = 0
I believe that this does satisfy the 0 vector requirement but am unsure of how to prove the addition requirement to show that it is a subspace.

Anyone familiar with vector spaces that can tell me if I'm on the right track and help me with the second part? Thanks.

 

[pka]

You are correct about part a).

For part b) you need to note something like the following:

\(\displaystyle \int\limits_0^1 {\alpha f + \beta g} = \int\limits_0^1 {\alpha f + } \int\limits_0^1 {\beta g} = \alpha \int\limits_0^1 {f + } \beta \int\limits_0^1 g\).

 

[jsbeckton]

Thanks, had something close to that. So part b satisfies both requirements and is a subspace correct?
btw, where do you get the integral and exponent symbols?

 

[pka]

At the top of this page is a tab for FORUM HELP
The mathematics is done with LaTeX.
There are several links on that tab.
If you use Windows, I suggest you download TeXaide.

 

[jsbeckton]

how do you add the formula typed in textaid to this forum, I tried and got a bunch of gibberish.

 

[pka]

Go ahead and post your "bunch of gibberish."
One of us can look at it and tell you what code to change.
Mostlikely you want to change \[ to \(\displaystyle .
And \] to \).

 

[jsbeckton]

Have a vector space problem that has me confused, it reads as follows:

Let \(\displaystyle S = \{ A \in M_2 (R):\det (A) = 0\}\)

a) Is the Zero vector from \(\displaystyle M_2 (R)\) in S?
A 2x2 0 matrix would have a determinate of 0 so I believe that it is in S.

b) Give an explicit example illistrating that S is not closed under matrix addition.
Any example that has real numbers and does not have a determinate of 0 will due, eg:

\(\displaystyle \matrix{
1 & 2 \cr
3 & 4 \cr\) ....... has a det(A) = -2 ....... not 0

c) Is S closed under scalar multiplication? Justify your anwser.
I believe that it is not closed under scalar multiplication since multiplying the matrix and the determinate by a scalar will not hold up but am not sure about that or how to justify it. Any help would be greatly appreciated. Thanks!

 

[pka]

Can you say "thank you" for the coding?

 

[jsbeckton]

I thanked you for your help, what are you talking about?

Thanks, had something close to that. So part b satisfies both requirements and is a subspace correct?
btw, where do you get the integral and exponent symbols?

I can make it work but I have to delete a bunch of nonsence. For example, I want to use the element symbol:

here is what happens if I just copy and paste it inside \(\displaystyle tags:
\(\displaystyle % MathType!MTEF!2!1!+-
% feqaeaartrvr0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l
% bbf9q8WrFfeuY-Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0-yr0R
% Yxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa
% caGaaeqabaaaamaaaOqaaiabgIGiodaa!3328!
$$
\in
$$\)

I know the proper coding is there at the end but where does the rest of that nonsence come from? Here is it after I delete all of the other characters:
\(\displaystyle \in\)\)

 

[jsbeckton]

Have a vector space problem that has me confused, it reads as follows:

Let \(\displaystyle S = \{ A \in M_2 (R):\det (A) = 0\}\)

a) Is the Zero vector from \(\displaystyle M_2 (R)\) in S?
A 2x2 0 matrix would have a determinate of 0 so I believe that it is in S.

b) Give an explicit example illistrating that S is not closed under matrix addition.
Any example that has real numbers and does not have a determinate of 0 will due, eg:

\(\displaystyle \matrix{
1 & 2 \cr
3 & 4 \cr\) ....... has a det(A) = -2 ....... not 0

c) Is S closed under scalar multiplication? Justify your anwser.
I believe that it is not closed under scalar multiplication since multiplying the matrix and the determinate by a scalar will not hold up but am not sure about that or how to justify it. Any help would be greatly appreciated. Thanks!

I'll move this to a new topic.

 

[pka]

I use MathType the larger version of TeXaide.
So this may no apply.

Under the Preferences Tab select Translators.
I use “TeX –LaTeX 2.09 later”.
Also UNCHECK boxes under the LaTeX.tdl .