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Math Help - Can somebody please help me prove this..thanks

  1. #1
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    Question Can somebody please help me prove this..thanks

    If the columns of A span Rn, then the columns of A form a basis of Rn.


    Also, I need to go the other way as well to make sure it is logically equivalent....I also need to show:

    If the columns of A form a basis of Rn, then the columns of A span Rn.

    I want to do a proof by contradiction...but not sure how.....This is sooooo confusing...thanks in advance
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  2. #2
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    Re: Can somebody please help me prove this..thanks

    It's a matter of definition. A basis of Rn is a set of vectors, such that every vector in Rn can be expressed as a linear combination of vectors in the set, and (another requirement) the vectors in the basis are linearly independent.

    Is there something in this you don't understand? Are you clear as to what linear independence is, and what a linear combination is? You should've been given a definition of "basis." Exactly what was it?

    It strikes me that the first statement you make, "if the columns of A span Rn,t hen the columns of A form a basis of Rn," is not strictly true. A could have more columns than there are elements in each column. The columns would then fail to form a basis because they wouldn't be linearly independent. Did you leave out part of the problem statement?

    The second statement, "if the columns of A form ... span Rn," seems to me to follow directly from the definition of basis.
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