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Math Help - dim null A-exam help wanted

  1. #1
    n22
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    dim null A-exam help wanted

    Hi,I am hoping someone can help me solve these questions:
    thanks
    let Ax=b be a consistent linear system,with b ≠0
    i) show that the set of solutions to the system is the set of vectors {xo+x such that x is an element of NullA, where xo is any solution to Ax=b
    ii) suppose that the dimension of Null(A) is zero. How many solutions are there to Ax=b?Explain..
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  2. #2
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    Re: dim null A-exam help wanted

    Quote Originally Posted by n22 View Post
    let Ax=b be a consistent linear system,with b ≠0
    i) show that the set of solutions to the system is the set of vectors {xo+x such that x is an element of NullA, where xo is any solution to Ax=b
    You need to show that two sets are equal. Do this by showing that each set contains the other. If x0 and x1 are two solutions of Ax = b, then what do you get by subtracting these equations, i.e., what can be said about x0 - x1?

    Quote Originally Posted by n22 View Post
    ii) suppose that the dimension of Null(A) is zero. How many solutions are there to Ax=b?Explain.
    If dim(Null(A)) = 0, then show (by contradiction) that A maps a basis into a basis. Therefore, b is a linear combination of the images of basis vectors under A. This gives at least one solution. Point i) implies that it is the only one.

    P.S. Don't make your posts in huge bold font. If you have less than perfect vision (as I do), then feel free to increase the font while you are editing and previewing the post, then remove the [SIZE=...]...[/SIZE] tags before submitting.
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