# Thread: dim null A-exam help wanted

1. ## 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..

2. ## Re: dim null A-exam help wanted

Originally Posted by n22
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?

Originally Posted by n22
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.

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