Question

Hi, I was wondering how would I approach this proof?

(i) (4 points) Let a E R&quot;, c &gt; 0, and
f (x) = a x+ cx x.
Show that f is coercive on Rn.
(You may assume linear functions and Euclidean norm are continuous.)
(ii) (6 points) Let C E Rnxn be a positive definite matrix, a E Rn, and
f (x) = a x+x Cx.
Show that f is coercive on Rn.
Hint: Use the Spectral Decomposition Theorem (see Theorem 2.4) from the lecture notes.
Also, as I mentioned in the class, positive semidefinite matrices and positive definite matrices are
symmetric (i.e., A = A ) by the way we defined them in the class. (See Definition 2.1. from the
lecture notes.)

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