Are binary quadratic and linear programs equivalent with the same constraints?

Question

If we consider the following two problems (1) and (2), are they equivalent? Because the values in $\bf x$ can only be either one or zero, the value of the objective function at the optimum $\bf x^*$ does not seem to change. Plus, since both problems are subjected to the same constraints, can I say that solving (1) also solves (2) or viceversa?

(1) $$\max {\bf x}^T {\bf c}$$ subject to $${\bf Ax} = {\bf b},$$ $${\bf x} = \{0, 1\}^M$$

(2) $$\max {\bf x}^T {\bf \mit diag\{\bf c\} \bf x}$$ subject to $${\bf Ax} = {\bf b},$$ $${\bf x} = \{0, 1\}^M$$

I would much apprecaite some comments and/or references.

Answer 1

Yes, of course they are exactly the same problem.

Answer 2

Yes.

$$x^Tdiag\{c\}x=\sum_{i=1}^M c_ix_i^2=\sum_{i=1}^M c_ix_i=x^Tc$$