About orthogonal complement in Gilbert Strang's "Linear Algebra and its Applications 2nd Edition"

Question

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?

Answer 1

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.

Answer 2

Let $A$ be an $m \times n$ matrix.
Let $R(A)$ be the row space of $A$.
Let $N(A)$ be the null space of $A$.
$\dim R(A) = r$, where $r$ is the rank of $A$.
$\dim N(A) = n - r$.
We can construct an orthonormal basis $v_1, \cdots, v_r$ of $R(A)$.
We can construct an orthonormal basis $w_1, \cdots, w_{n-r}$ of $N(A)$.
Then, the vectors $v_1, \cdots, v_r, w_1, \cdots, w_{n-r}$ are orthonormal because $R(A)$ and $N(A)$ are orthogonal.
So, the vectors $v_1, \cdots, v_r, w_1, \cdots, w_{n-r}$ are linearly independent.
Because $\#\{v_1, \cdots, v_r, w_1, \cdots, w_{n-r}\} = n$, $v_1, \cdots, v_r, w_1, \cdots, w_{n-r}$ are an orthonormal basis of $\mathbb{R}^n$.

Let $x \in \mathbb{R}^n$.

Then we can write as $x = a_1 v_1 + \cdots, a_r v_r + b_1 w_1 + \cdots + b_{n-r} w_{n-r}$, where $a_i, b_j \in \mathbb{R}$.
$a_1 v_1 + \cdots, a_r v_r \in R(A)$ and $b_1 w_1 + \cdots + b_{n-r} w_{n-r} \in N(A)$.