Results 1 to 5 of 5

Math Help - Is R^3 a subspace of C^3

  1. #1
    Member
    Joined
    Feb 2008
    Posts
    184

    Is R^3 a subspace of C^3

    HELLO,

    Is a subspace of {C^3}?

    C=complex numbers.

    Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Gamma's Avatar
    Joined
    Dec 2008
    From
    Iowa City, IA
    Posts
    517
    Is it closed under addition? yes
    Is it closed under scalar multiplication from the field C? no
    <1,1,1> \in \mathbb{R}^3 but i<1,1,1>=<i,i,i> \notin \mathbb{R}^3
    Not a subfield.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Feb 2008
    Posts
    184
    How do i prove closed under addition condition.
    you took i from complex vector space C. Can you do

    ((1+i),1,1) \notin \mathbb{R}^3
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member Gamma's Avatar
    Joined
    Dec 2008
    From
    Iowa City, IA
    Posts
    517
    well vector addition is just done componentwise, so if you add two real valued vectors, their sum is clearly still real valued since the real numbers are closed under addition. I mean (a,b,c)+(d,e,f)=(a+d,b+e,c+f) where everything is necessarily real numbers, so its closed under addition. But to be a subspace it also needs to be closed under scalar multiplication, and it is not, this is enough to prove it is not a subspace.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    I think it depends on the field.

    For example: Every field is a one dimentional vector space over itself.

    This implies that \mathbb{C} has A basis \mathcal{B}=\{ 1\} when the scalers are take as complex numbers.

    On the other hand \mathbb{C} has A basis \mathcal{B'}=\{ 1,i\} when the scalers are taken as real numbers. It is a two dimentional vectors space over \mathbb{R}

    Now we could view \mathbb{R} as a closed subspace of the 2nd, but not the first.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Why is this set a subspace ??
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: July 3rd 2010, 09:10 PM
  2. Subspace spanned by subspace
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: February 9th 2010, 07:47 PM
  3. Subspace
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: February 9th 2010, 03:08 AM
  4. How many subspace in F2?
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: January 11th 2009, 12:35 PM
  5. Subspace of R3
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: December 14th 2008, 07:36 AM

Search Tags


/mathhelpforum @mathhelpforum