Results 1 to 7 of 7

Math Help - dimention laws

  1. #1
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401

    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

    ??
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1

    Re: dimention laws

    Quote Originally Posted by transgalactic View Post
    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.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401

    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?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Lord of certain Rings
    Isomorphism's Avatar
    Joined
    Dec 2007
    From
    IISc, Bangalore
    Posts
    1,465
    Thanks
    6

    Re: dimention laws

    Quote Originally Posted by transgalactic View Post
    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 \text{dim}(\text{sp}\{v1,w2\})=2. The plan is to show that \text{dim}(\text{sp}\{v1,w2\})\le 2 and \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

    ??
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401

    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 .
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor Also sprach Zarathustra's Avatar
    Joined
    Dec 2009
    From
    Russia
    Posts
    1,506
    Thanks
    1

    Re: dimention laws

    Quote Originally Posted by transgalactic View Post
    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 .

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

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

    a_1v_1+a_2v_2=b_1w_1+b_2w_2

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

    t_1v_1+t_2v_2=0

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

    0=t_1v_1+t_2v_2=b_1w_1+b_2w_2

    or:

    b_1w_1+b_2w_2=0

    So w_1 and w_2 are independent.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor
    Joined
    Nov 2008
    Posts
    1,401

    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
    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
    a_1v_1+a_2v_2=b_1w_1+b_2w_2
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. dimention of kernel
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: June 29th 2011, 10:33 AM
  2. dimention of group question
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: June 22nd 2011, 03:01 PM
  3. Index Laws
    Posted in the Algebra Forum
    Replies: 1
    Last Post: November 22nd 2008, 06:51 PM
  4. laws
    Posted in the Algebra Forum
    Replies: 1
    Last Post: September 25th 2008, 07:23 AM
  5. Log Laws
    Posted in the Math Topics Forum
    Replies: 1
    Last Post: September 1st 2008, 04:45 AM

Search Tags


/mathhelpforum @mathhelpforum