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Math Help - Basis in Linear Algebra

  1. #1
    Member akhayoon's Avatar
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    Basis in Linear Algebra

    what exactly is a basis?

    all these textbook explantions are so confusing to me and usually have all kinds of symbols that I have to decipher making it even harder to read let alone understand.
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  2. #2
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    Quote Originally Posted by akhayoon View Post
    what exactly is a basis?
    .
    Let R be the real numbers and V be a vector space over R. The set of vectors {v1,v2,...,vn} is said to form a basis when any vector in V can be expressed* in terms of v_1,v_2,...,v_n AND in only one way.

    For example, \bold{i},\bold{j},\bold{k} is a basis for V=\mathbb{R}^3 over \mathbb{R}. Because anything in V can be expressed in terms of those.


    *)By "expressed" I mean as a linear combination of those vectors.
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  3. #3
    Member akhayoon's Avatar
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    but I thought if they were linear combinations that means they are linearly dependent and therefore are not a basis of that subplace or whatever they call it...lol
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  4. #4
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    but I thought if they were linear combinations that means they are linearly dependent and therefore are not a basis of that subplace or whatever they call it...lol
    If any vectors in a set are linear combinations of each other then they are linearly dependent and not a basis.
    If a set is a basis of a space then every vector in the space is a linear combination of the vectors in the basis.
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  5. #5
    MHF Contributor kalagota's Avatar
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    i think, TPH missed an important point.. the set must be linearly independent..
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  6. #6
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    Quote Originally Posted by kalagota View Post
    i think, TPH missed an important point.. the set must be linearly independent..
    No! I said the representation is unique.
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  7. #7
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by ThePerfectHacker View Post
    No! I said the representation is unique.
    hehehe.

    you already included a theorem in the definition..
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