## Positive-definiteness

Dear Dr. Kim,

Could you show me how to determine whether it is positive definite for homework 8 question 1c please.
It will be very grateful if you can show me in your course blog.

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The question takes the vector space $V$ to be the span of $\{(1,1,1)^t, (1,2,3)^t\}$ in $C^3$, and the Hermitian form to be
$\langle v, w \rangle=v^t\bar{w}.$
We need to see if $\langle v,v\rangle >0$ for all non-zero $v=(v_1,v_2,v_3)^t \in V$. But
$\langle v, v\rangle =v^t\bar{v}=|v_1|^2+|v_2|^2+|v_3|^2>0$
as long as some $v_i$ is non-zero. So the form is positive definite. Notice that this form is actually defined and is positive definite on all of $C^3$. In general if you have a Hermitian form on a $C$-vectors space or a symmetric bilinear form on an $R$-vector space, if the original form is positive definite, then its restriction to any subspace is also positive definite, simply by exmaining the definition of positive-definiteness.