TITLE: Orthogonality of row space and null space
QUESTION [2 upvotes]: I know that according to the Fundamental theorem of linear algebra the row space and the null space are orthogonal, but I don't really understand why.  Could someone give an intuitive explanation of why this is with maybe some examples from $\mathbb{R}^2$ or $\mathbb{R}^3$ with the standard Euclidean inner product?

REPLY [4 votes]: The row space is the set of $A^Tx$ for every vector $x$, the null space is the set of vectors $y$ such that $Ay=0$. The scalar product between a vector in the row space and a vector in the null space is $\langle y,A^Tx\rangle=y^T(A^Tx)=x^T(Ay)=x^T0=0$. The second equality follows from the fact that $y^TA^Tx$ has size $1\times 1$, hence is equal to its transpose $x^TAy$.
{"set_name": "stack_exchange", "score": 2, "question_id": 213516}