What is the correct way to connect the subspaces of a matrix to transpose vectors in the method of least squares?

Question

I'm studying linear algebra alone and having a hard time dealing with transpose vectors and their subspaces. The summary of what I saw is as follows:

• The row space of $A$ is the column space of $A^T$ and vice versa.
• The null space of $A=\{x\mid Ax=0,\ x^TA^T=0\}$ is the left null space of $A^T=\{x\mid A^Tx=0,\ x^TA=0\}$ and vice versa.
• If a solution of $Ax=b$ or $x^TA^T=b^T$ exists, $b$ is an element of column space of $A$ and row space of $A^T$.
• If a solution of $A^Tx=b$ or $x^TA=b^T$ exists, $b$ is an element of row space of $A$ and column space of $A^T$.
• The vector $x$ which minimizes $|b-Ax|$ causes $b-Ax$ to belong to the left null space of $A$. Therefore projection $b$ to column space of $A$ is $A\hat{x}=A((A^TA)^{-1}A^Tb)$.

My questions are:

• If $b$ is in column space of $A$, is $b^T$ in row space of $A$?
• Is the following statement correct?

  The vector $x$ which minimizes $|b-A^Tx|$ causes $b-A^Tx$ to belong to the null space of $A$. Therefore projection $b$ to row space of $A$ is $A^T\hat{x}=A^T((AA^T)^{-1}Ab)$.

• How can I use $x^TA^T=b^T$ to describe the same as above? Does $b^T-x^TA^T$ belong to left null space of $A$? Which should I use, $A^Tx=b$ or $x^TA=b^T$?

Answer 1

• Q1: It's incorrect, since $b\in C(A)$ ($C$ for column space, $R$ for row space) $\Rightarrow b^{T}\in R(A^{T})$, which is generally different from $R(A)$. (Or to be brief, $b$ contains only the column information which isn't affected after transposition.)
• Q2: This is true by substituting $A$ with $A^T$ and $C(A^T)$ with $R(A)$. (Although the wordings in this question should be like $b-Ax\in ...$, rather than $|b-Ax|\in ...$)
• Q3: If $|b^T-x^TA^T|=|(b-Ax)^T|=|b-Ax|$ is minimalized, then $b-Ax$ belongs to the left null space of A, thus $A^T(b-Ax)=0\Rightarrow (b^T-A^Tx^T)A=0$, so this statement is also true.