Consider the matrix `A = [[2, 1, 3],[4, 1, 4]]` . Show that `N(A)` is perpendicular to `R(A)` , where `N(A)` stands for the null space of `A` and `R(A)` the row space of `A.`

Expert Answers
degeneratecircle eNotes educator| Certified Educator

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.