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Math Help - 2-dimensional subspace U of C^3.

  1. #1
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    2-dimensional subspace U of C^3.

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

    thanks
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  2. #2
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    \mathbb{C} is two dimensional since \mathbb{C} = \{a+ ib \colon a,b\in\mathbb{R}\}, which means that \mathbb{C} is a vector space with two \mathbb{R} entries.

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

    Now using the definition of subspace:
    Since 0 \in \mathbb{C}
    and \mathbb{C} is closed under addition and scalar multiplication \mathbb{C} = U is a two dimensional subspace of \mathbb{C}^{3}
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    Thanks for your help. Now how do i
    write a basis for this subspace.
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    Well a basis is any linearly independent that spans the vector space. In the case of \mathbb{C}, no real constants a,b, will make a + ib = 0 unless a=b=0. So any ordered pair of real numbers will constitute a basis for \mathbb{C}
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    MHF Contributor Bruno J.'s Avatar
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    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 C is the field \mathbb{C}, then it is unclear over what field you are considering \mathbb{C}^3 to be a vector space. For instance it's a 6-dimensional vector space over \mathbb{R}, but a 3-dimensional one over \mathbb{C}. You always need to specify the underlying field. Haven, above, assumed that the vector space is over \mathbb{R}. Check your question!
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  6. #6
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    Question is:
    Give an example of a 2-dimensional subspace of
    \mathbb{C}^3

    thanks
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    Noting Bruno's comment on fields. If we assume \mathbb{C}^{3} is an \mathbb{R} vector space, then my above answer is correct. However if we view \mathbb{C}^{3} as a \mathbb{C} vector space. Then an example of a two-dimensional vector space would be \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 \mathbb{C}^{3} as an  \mathbb{R} vector space, is probably the desired one.
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