I am reading Gilbert Strang's "Linear Algebra and its Applications 2nd Edition".
On p.137(3.4 The PSEUDOINVERSE AND THE SINGULAR VALUE DECOMPOSITION), he wrote "any vector can be split into two perpendicular pieces, its projection onto the row space and its projection onto the nullspace".
I cannot find the proof of the above fact in his book.
I think he didn't prove the above fact in his book, but he used the fact in his book.
By the way, I think he proved the row space and nullspace of $A$ are orthogonal complements in $\mathbb{R}^n$.
Am I correct or not?
Best Answer
This is because if $Ax = 0$ for $x$ in the nullspace, then each component of $Ax$ must be zero. But the components $[Ax]_j = 0$ corresponds to the dot product of the $j$-th row and $x$, which makes $x$ perpendicular to the entire row space. Therefore, the row space and the null space of $A$ are perpendicular.