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Math Help - Linear algebra Question check

  1. #1
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    Linear algebra Question check

    Question: Let B1;B2 be given column vectors in Rm and suppose that the set {B1,B2} is independent. Suppose that A is an mxn matrix and that AX1 = B1, AX2 = B2 for some column vectors X1;X2 in Rn. Show that {X1,X2} is independent.

    Thought: well you need to show that c1x1 + c2x2 = 0 --> c1 = c2 = 0. Suppose that c1x1 + c2x2 = 0. Then A(c1x2 + c2x2) = c1B1 + c2B2 = 0,because A(0) = 0 for any matrix A.
    since{B1, B2} is independent, c1 = c2 = 0. Thus {x1,x2} is independent as well.


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  2. #2
    Senior Member Tinyboss's Avatar
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    For a set of two vectors, being linearly dependent is equivalent to one being a scalar multiple of the other. Remember matrix multiplication is a linear map. So for any matrix M and vector v (of compatible sizes) and scalar a, you have M(av) = a(Mv). So, if X1 and X2 were dependent, then X1=aX2 for some a. Can you finish from there?
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