# dimention laws

• Jun 28th 2011, 03:07 AM
transgalactic
dimention laws
v1,v2,w1,w2 are independant vectors in space V
so Sp{v1,v2)=Sp{w1,w2} and {v1,w2} independant.
prove that {v1,v2} is independant.

before solving this question i got some other questions.

first question:
if {v1,w2} is independant what can i say about the dimention of sp{v1,w2}

i was told that dim sp{v1,w2} >=2

how can it be that two vector can span a dimention bigger then 2?
??

second question
beacause there where only two vectors sp{v1,w2} <=2

so in sum of both thing we get dim sp{v1,w2}=2

cant understand the reason why on the one hand its bigger then 2
and on the other one its smaller then 2

??
• Jun 28th 2011, 03:28 AM
Also sprach Zarathustra
Re: dimention laws
Quote:

Originally Posted by transgalactic
v1,v2,w1,w2 are independant vectors in space V
so Sp{v1,v2)=Sp{w1,w2} and {v1,w2} independant.
prove that {v1,v2} is independant.

before solving this question i got some other questions.

first question:
if {v1,w2} is independant what can i say about the dimention of sp{v1,w2}

i was told that dim sp{v1,w2} >=2

how can it be that two vector can span a dimention bigger then 2?
??

second question
beacause there where only two vectors sp{v1,w2} <=2

so in sum of both thing we get dim sp{v1,w2}=2

cant understand the reason why on the one hand its bigger then 2
and on the other one its smaller then 2

??

By definition {v1,w2} is independent and: U=sp{v1,w2} then dim(U)=2.
• Jun 28th 2011, 04:16 AM
transgalactic
Re: dimention laws
so this game of summing too innequalities
is irrelevant.
and i can say that if i have 2 intependant vectors {v1,v2}
then dim sp{v1,v2}=2

ok now i will try and solve the original question.

v1,v2,w1,w2 are independant vectors in space V
so Sp{v1,v2)=Sp{w1,w2} and {v1,w2} independant.
prove that {v1,v2} is independant.

Sp{v1,v2,w1,w2}=sp{v1,v2}+sp{w1,w2}

we are given that {v1,v2} is independant so dim Sp{v1,v2,w1,w2} >=2

what to do now?
• Jun 28th 2011, 12:23 PM
Isomorphism
Re: dimention laws
Quote:

Originally Posted by transgalactic
v1,v2,w1,w2 are independant vectors in space V
so Sp{v1,v2)=Sp{w1,w2} and {v1,w2} independant.
prove that {v1,v2} is independant.

It is given that v1,v2, w1 and w2 are independent.
If you have proved that every subset of an independent set is independent, then there is nothing remaining to prove.

Quote:

before solving this question i got some other questions.

first question:
if {v1,w2} is independant what can i say about the dimention of sp{v1,w2}
The below argument is proving that $\displaystyle \text{dim}(\text{sp}\{v1,w2\})=2$. The plan is to show that $\displaystyle \text{dim}(\text{sp}\{v1,w2\})\le 2$ and $\displaystyle \text{dim}(\text{sp}\{v1,w2\})\ge 2$ are both true.

Quote:

i was told that dim sp{v1,w2} >=2

how can it be that two vector can span a dimention bigger then 2?
??

ANSWER: Since the subspace, span{v1, w2} has at least two independent vectors (namely v1 and w2), dim sp{v1,w2} >=2. The idea is that a basis is a maximal independent set and dimension is the cardinality of a basis.

second question
beacause there where only two vectors sp{v1,w2} <=2

ANSWER: Because span{v1,w2} can be spanned by v1 and w2, i.e two vectors and a basis is a minimal spanning set. The number of elements in a basis will be less than or equal to number of elements in the spanning set {v1,w2}.
so in sum of both thing we get dim sp{v1,w2}=2

cant understand the reason why on the one hand its bigger then 2
and on the other one its smaller then 2

??
• Jun 29th 2011, 03:40 AM
transgalactic
Re: dimention laws
sorry i am not given that
v1,v2,w1,w2 are independant vectors

its a typo

so Sp{v1,v2)=Sp{w1,w2} and {v1,w2} independant.
prove that {v1,v2} is independant.

Sp{v1,v2,w1,w2}=sp{v1,v2}+sp{w1,w2}

we are given that {v1,v2} is independant so dim Sp{v1,v2,w1,w2} >=2

what to do now? Edit Post Reply Reply With Quote Thanks .
• Jun 29th 2011, 04:17 AM
Also sprach Zarathustra
Re: dimention laws
Quote:

Originally Posted by transgalactic
sorry i am not given that
v1,v2,w1,w2 are independant vectors

its a typo

so Sp{v1,v2)=Sp{w1,w2} and {v1,w2} independant.
prove that {v1,v2} is independant.

Sp{v1,v2,w1,w2}=sp{v1,v2}+sp{w1,w2}

we are given that {v1,v2} is independant so dim Sp{v1,v2,w1,w2} >=2

what to do now? Edit Post Reply Reply With Quote Thanks .

$\displaystyle Sp\{v_1,v_2\}=Sp\{w_1,w_2\}$

So there exist $\displaystyle a_1,a_2,b_1,b_2 \in \mathbb{F}$ so:

$\displaystyle a_1v_1+a_2v_2=b_1w_1+b_2w_2$

But, $\displaystyle v_1, v_2$ are independent, so there exist $\displaystyle t_1, t_2 \in \mathbb{F}$ so that:

$\displaystyle t_1v_1+t_2v_2=0$

If e choose $\displaystyle a_1=t_1$ and $\displaystyle a_2=t_2$ we will get:

$\displaystyle 0=t_1v_1+t_2v_2=b_1w_1+b_2w_2$

or:

$\displaystyle b_1w_1+b_2w_2=0$

So $\displaystyle w_1$ and $\displaystyle w_2$ are independent.
• Jun 29th 2011, 05:39 AM
transgalactic
Re: dimention laws
we are not given that {v1,v2} is independant
so i dissagree with what you did afterwards
we were given that {v1,w2}

so
$\displaystyle av_1+bv_2=0$ only if a=b=0

from the span equality each vector in both groups could be found by lenear combination
$\displaystyle a_1v_1+a_2v_2=b_1w_1+b_2w_2$