1. The problem states that $x_{opt}$ is the least squares solution to $Ax = b$ and asks about the nature of the error vector $e = Ax_{opt} - b$. 2. Recall that the least squares solution $x_{opt}$ minimizes the squared error $\|Ax - b\|^2$. 3. The vector $Ax_{opt}$ is the projection of $b$ onto the column space of $A$ because the least squares solution projects $b$ onto the space spanned by the columns of $A$. 4. Therefore, the error vector $e = Ax_{opt} - b$ is the difference between the projection and $b$, which means $e$ is orthogonal to the column space of $A$. 5. The orthogonal complement of the column space of $A$ is the null space of $A^T$. 6. Hence, the error vector $e$ lies in the null space of $A^T$ and is the projection of $b$ onto the null space of $A^T$. Final answer: The error vector $e$ is the projection of $b$ onto the null space of $A^T$.