Using the Hessian, prove that affine scalar field is convex

Question

Using the Hessian, prove that the affine scalar field $f : \Bbb R^n \to \Bbb R$ defined by $$f(x) = A^T x + b$$ where $A \in \Bbb R^n$ and $b \in \Bbb R$, is convex.

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My intuition on this problem is to apply the theorem that if $f(x)$ is twice continuously differentiable then $f(x)$ is convex if and only if the Hessian is positive semidefinite, but I don't know how to find the hessian of this function and show that it is positive semidefinite.

Answer 1

If $A^T=(a_1,a_2,...,a_n)$ and $x=(x_1,...,x_n)^T$, then

$$f(x)=a_1x_1+...+a_nx_n+b.$$

Hence

$$f_{x_j}(x)=a_j$$

for $j=1,...,n.$

Conclusion: the hessian of this function $=?$