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Math Help - Linearly independent coordinate vectors help please?

  1. #1
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    Linearly independent coordinate vectors help please?

    I'm having trouble with proofs in linear algebra. I don't know where to start and how to go about the proofs.Any help would be appreciated.

    (1) Let S be a basis for an n-dimensional vector space. Show that a set of vectors {v1,v2,...,vr} form a linearly independent set if and only if their coordinate vectors with respect to S namely (v1)s, (v2)s , ..., (vr)s are linearly independent in R^n.
    I know that S being a basis for an n-dimensional vector space means that it is linearly independent and spans R^n but I don't know how to relate this to the proof.

    (2)Show that if {v1,v2,...vr} span V, then (v1)s ,(v2)s ,..., (vr)s span R^n and conversely.
    (This is a follow up from the previous question) I don't know where to start.

    Any input would help. Thanks in advance
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  2. #2
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    Quote Originally Posted by chocaholic View Post
    I'm having trouble with proofs in linear algebra. I don't know where to start and how to go about the proofs.Any help would be appreciated.

    (1) Let S be a basis for an n-dimensional vector space. Show that a set of vectors {v1,v2,...,vr} form a linearly independent set if and only if their coordinate vectors with respect to S namely (v1)s, (v2)s , ..., (vr)s are linearly independent in R^n.
    I know that S being a basis for an n-dimensional vector space means that it is linearly independent and spans R^n but I don't know how to relate this to the proof.
    The definition of "linearly independent" is that the only way we can have a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_rv_r= 0 is to have a_1= a_2= \cdot\cdot\cdot= a_r= 0. Write the equation a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_rv_r= 0 in terms of (v_1)s, (v_2)s, ..., (v_3)s to get an equation in R^n.


    (2)Show that if {v1,v2,...vr} span V, then (v1)s ,(v2)s ,..., (vr)s span R^n and conversely.
    (This is a follow up from the previous question) I don't know where to start.

    Any input would help. Thanks in advance
    The definition of "span V" is that if v is any vector in V, then there exist numbers a_1, a_2, ..., a_r such that a_1v_1+ a_2v_2+ \cdot\cdot\cdot+ a_rv_r= y. Again, write that equation in terms of (v_1)s, (v_2)s, ..., (v_r)s.

    Mathematics definitions are working definitions- learn the precise words of definitions and use them in proofs.
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  3. #3
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    Re: Linearly independent coordinate vectors help please?

    Added: The "standard basis" in R^n is the set of vectors, <1, 0, 0, ..., 0>, <0, 1, 0, ..., 0> ..., <0, 0, 0,..., 1>. So write whatever vectors you are given in that form:
    if v_1= <v_{11}, v_{12}, ..., v_{1n}>, v_2= <v_{21}, v_{22}, ..., v_{2n}>, ..., v_n= <v_{n1}, v_{n2}, ..., v_{nn}>.

    Then a_1v_1+ a_2v_2+ ....+ a_nv_n= <a_1v_{11}+ a_2v_{21}+...+ a_nv_{n1}, a_1v_{12}+ a_2v_22+ ....+ a_nv{n2}, ..., a_1v_{1n}+ a_2v_{2n}+ ...+ a_nv_{nn}>.
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