1. ## 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

??

2. ## Re: dimention laws

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.

3. ## 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?

4. ## Re: dimention laws

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.

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.

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

??

5. ## 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 .

6. ## Re: dimention laws

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.

7. ## 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
$\displaystyle a_1v_1+a_2v_2=b_1w_1+b_2w_2$