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The residual vector $\vec{r} = A\vec{x}^* - \vec{b}$ is orthogonal to the column space $C(A)$ because $A\vec{x}^*$ is the orthogonal projection of $\vec{b}$ onto $C(A)$. Orthogonality to $C(A)$ means $\vec{r}$ is in the orthogonal complement $C(A)^\perp$.

Conditions: $A\vec{x}^*$ is the orthogonal projection of $\vec{b}$ onto $C(A)$; $C(A)^\perp = N(A^T)$ (Fundamental Theorem of Linear Algebra); Matrix multiplication distributes over subtraction

The residual vector $\vec{r} = A\vec{x}^* - \vec{b}$ is orthogonal to the column space $C(A)$ because $A\vec{x}^*$ is the orthogonal projection of $\vec{b}$ onto $C(A)$. Orthogonality to $C(A)$ means $\vec{r}$ is in the orthogonal complement $C(A)^\perp$.

Conditions: $A\vec{x}^*$ is the orthogonal projection of $\vec{b}$ onto $C(A)$; $C(A)^\perp = N(A^T)$ (Fundamental Theorem of Linear Algebra); Matrix multiplication distributes over subtraction