Suppose A is an mxn matrix of real numbers and x in an nx1 column vector....

Suppose A is an mxn matrix of real numbers and x in an nx1 column vector.

a.) suppose Ax=0. Show that A^TAx=0.

b.)Suppose A^TAx=0. show Ax=0.

c.) by part a and b, we can conclude that Nul(A) = Nul(A^TA), and thus dim(Nul A) = dim(Nul(A^TA)), and thus nullity(A) = nullity(A^TA). prove the columns of A are linearly independent iff A^TA is invertible.

Expert Solution

Colby Messinger answered 2 years ago

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