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Consider the matrix `A = [[2, 1, 3],[4, 1, 4]]` . Show that `N(A)` is perpendicular...
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High School Teacher
Instead of showing it for this particular case, it is actually less work to prove that the null space and row space are orthogonal complements (perpendicular) for any matrix `A.` To do this, suppose that `x in N(A),` so that `Ax=0.` Also, the row space of `A` is the same as the column space of `A^T.` Thus `y in R(A)` if and only if `A^T bary=y` for some vector `bary.` We must show that `x*y=0,` and we can use the fact that `x*y=x^Ty.` We get
`x^Ty=x^T(A^Tbary)=(x^TA^T)bary=(Ax)^Tbary=0^Tbary=0,` ``which is what we needed to show. See the link for a slightly different explanation.
Posted by degeneratecircle on April 13, 2013 at 4:38 AM (Answer #1)
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