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.`
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.