dimention of group question

there is

v1,..,vk,u,w vectors on space V

the group {v1,..,vk,u} is independant

and

$\displaystyle Sp\{v1,..,vk\}\cap Sp\{u,w\}\neq\{0\}$

find the dimention of sp{v1..vk,u,w}

?

if {v1,..,vk,u} is independant then its span dimention is k+1

and span so {v1,..,vk} independat to and dim sp{v1,..,vk}=k

the dimention of sp{v1..vk,u,w} is

dim(sp{v1..vk,u,w})=dim(sp{v1..vk}and sp{u,w}})=dim(sp{v1..vk}+sp{u,w})=dim(sp{v1..vk})+ dim(sp{u,w})-dim(sp{v1..vk} intersect sp{u,w}})

if v1..vk,u is independant then v1,..,vk is independant too so dim (sp{v1..vk})=k

now regardingthe other two members i dont know

Re: dimention of group question

Let $\displaystyle x$ be a nonzero vector in

$\displaystyle \text{span}\{v_1,\cdots,v_k\}\cap\text{span}\{u,w\ }$.

Then there are constants $\displaystyle a_1,\cdots,a_k$ and $\displaystyle b,c$ such that $\displaystyle a_1v_1+\cdots+a_kv_k=x=bu+cw$ and hence

$\displaystyle w=c^{-1}a_1v_1+\cdots+c^{-1}a_kv_k-c^{-1}bu$.

So $\displaystyle w\in\text{span}\{v_1,\cdots,v_k,u\}$ which means $\displaystyle \dim\text{span}\{v_1,\cdots,v_k,u,w\}=\dim\text{sp an}\{v_1,\cdots,v_k,u\}=k+1$.

Re: dimention of group question

ok so we need to find

dim(sp{v1,..vk,u,w})=dim(sp{v1,..,vk,u}+sp{w})

dim(sp{v1,..,vk,u}+sp{w})=dim(sp{v1,..,vk,u})+dim( sp{w})-$\displaystyle dim(sp{v1,..,vk,u}\cap sp{w})$=k+1

beacuse $\displaystyle dim(sp({v1,..,vk,u}}) \cap sp{w})=0$

so

dim(sp{v1,..vk,u,w})=k+1

correct?

Re: dimention of group question

Quote:

Originally Posted by

**transgalactic** ok so we need to find

dim(sp{v1,..vk,u,w})=dim(sp{v1,..,vk,u}+sp{w})

dim(sp{v1,..,vk,u}+sp{w})=dim(sp{v1,..,vk,u})+dim( sp{w})-$\displaystyle dim(sp{v1,..,vk,u}\cap sp{w})$=k+1

beacuse dim(sp{v1,..,vk,u}\cap sp{w})=0

so

dim(sp{v1,..vk,u,w})=k+1

correct?

This is extremely unreadable. When you write out proofs in mathematics, you need to make sure that everyone can understand what you are trying to say. We aren't in your head; all we see is what you have written down, so if we can't interpret that, then we are out of luck.

Anyway, **hatsoff** has given the complete proof, with no steps missing.

Re: dimention of group question

hatsoff proved a very important step

it showed that i divided the group in the wring way

i made some mistakes in latex

but its only about one formula for the dimention of the sum of spans

here is the edited:

ok so we need to find

dim(sp{v1,..vk,u,w})=dim(sp{v1,..,vk,u}+sp{w})

dim(sp{v1,..,vk,u}+sp{w})=dim(sp{v1,..,vk,u})+dim( sp{w})-$\displaystyle dim(sp{v1,..,vk,u}\cap sp{w})$=k+1

beacuse $\displaystyle dim(sp({v1,..,vk,u}}) \cap sp{w})=0$

so

dim(sp{v1,..vk,u,w})=k+1

correct?

Re: dimention of group question

Quote:

Originally Posted by

**transgalactic** hatsoff proved a very important step

it showed that i divided the group in the wring way

i made some mistakes in latex

but its only about one formula for the dimention of the sum of spans

here is the edited:

ok so we need to find

dim(sp{v1,..vk,u,w})=dim(sp{v1,..,vk,u}+sp{w})

dim(sp{v1,..,vk,u}+sp{w})=dim(sp{v1,..,vk,u})+dim( sp{w})-$\displaystyle dim(sp{v1,..,vk,u}\cap sp{w})$=k+1

beacuse $\displaystyle dim(sp({v1,..,vk,u}}) \cap sp{w})=0$

so

dim(sp{v1,..vk,u,w})=k+1

correct?

1. How do you get $\displaystyle dim(sp({v1,..,vk,u}}) \cap sp{w})=0$?

2. If (1) were true, then

dim(sp{v1,..,vk,u}+sp{w})=dim(sp{v1,..,vk,u})+dim( sp{w})-$\displaystyle dim(sp{v1,..,vk,u}\cap sp{w})$=dim(sp{v1,..,vk,u})+dim(sp{w})-0=(k+1)+1-0=k+2, NOT k+1.

I'll say again: **hatsoff** has already given the COMPLETE proof.

Re: dimention of group question

oohh yes i got it

soryy i am a little slow on the comprehetion side in this hour

thanks :)