Results 1 to 7 of 7

Thread: 2-dimensional subspace U of C^3.

  1. #1
    Member
    Joined
    Feb 2008
    Posts
    184

    2-dimensional subspace U of C^3.

    Can you please Give me an example of a 2-dimensional subspace U of C^3.

    thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member Haven's Avatar
    Joined
    Jul 2009
    Posts
    197
    Thanks
    8
    $\displaystyle \mathbb{C}$ is two dimensional since $\displaystyle \mathbb{C} = \{a+ ib \colon a,b\in\mathbb{R}\}$, which means that $\displaystyle \mathbb{C} $ is a vector space with two $\displaystyle \mathbb{R}$ entries.

    It's clear that $\displaystyle \mathbb{C} \subset \mathbb{C} ^ {3}$

    Now using the definition of subspace:
    Since $\displaystyle 0 \in \mathbb{C}$
    and $\displaystyle \mathbb{C}$ is closed under addition and scalar multiplication $\displaystyle \mathbb{C} = U$ is a two dimensional subspace of $\displaystyle \mathbb{C}^{3}$
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Feb 2008
    Posts
    184
    Thanks for your help. Now how do i
    write a basis for this subspace.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member Haven's Avatar
    Joined
    Jul 2009
    Posts
    197
    Thanks
    8
    Well a basis is any linearly independent that spans the vector space. In the case of $\displaystyle \mathbb{C}$, no real constants a,b, will make $\displaystyle a + ib = 0$ unless a=b=0. So any ordered pair of real numbers will constitute a basis for $\displaystyle \mathbb{C}$
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    Quote Originally Posted by charikaar View Post
    Can you please Give me an example of a 2-dimensional subspace U of C^3.

    thanks
    Your question is imprecise. First, if $\displaystyle C$ is the field $\displaystyle \mathbb{C}$, then it is unclear over what field you are considering $\displaystyle \mathbb{C}^3$ to be a vector space. For instance it's a 6-dimensional vector space over $\displaystyle \mathbb{R}$, but a 3-dimensional one over $\displaystyle \mathbb{C}$. You always need to specify the underlying field. Haven, above, assumed that the vector space is over $\displaystyle \mathbb{R}$. Check your question!
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Feb 2008
    Posts
    184
    Question is:
    Give an example of a 2-dimensional subspace of
    $\displaystyle \mathbb{C}^3$

    thanks
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member Haven's Avatar
    Joined
    Jul 2009
    Posts
    197
    Thanks
    8
    Noting Bruno's comment on fields. If we assume $\displaystyle \mathbb{C}^{3}$ is an $\displaystyle \mathbb{R}$ vector space, then my above answer is correct. However if we view $\displaystyle \mathbb{C}^{3}$ as a $\displaystyle \mathbb{C}$ vector space. Then an example of a two-dimensional vector space would be $\displaystyle \mathbb{C}^{2}$

    I assume based on the vagueness of the question that you are in a Linear Algebra class and you have little or no understanding of what a field is (in the context of the class). So the interpretation of $\displaystyle \mathbb{C}^{3}$ as an$\displaystyle \mathbb{R}$ vector space, is probably the desired one.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Let Z be a proper subspace of an n-dimensional vector space X
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: Mar 29th 2011, 11:56 AM
  2. Replies: 1
    Last Post: Feb 25th 2010, 01:15 AM
  3. Subspace spanned by subspace
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: Feb 9th 2010, 07:47 PM
  4. one-dimensional subspace of R^3
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: Nov 18th 2009, 12:09 PM
  5. A one dimensional...
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Jul 21st 2006, 11:18 PM

Search Tags


/mathhelpforum @mathhelpforum