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\title{MAT 247 \\ Assignment 5 \\ Due Thursday March 3}



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\begin{enumerate}
\item (Axler 7.19) Suppose that $ T $ is a positive operator on an inner product space $ V $.  Prove that $ T $ is invertible if and only if 
$$
\langle T v, v \rangle > 0 
$$
for all $ v \in V \setminus \{0\} $.

\item (Axler 7.20) Prove or disprove: the identity operator on $ \mathbb{R}^2 $ has infinitely many square roots.

\item (Axler 7.23) Define a linear operator $ T $ on $ \mathbb{R}^3 $ by $T(a_1, a_2, a_3) = (a_3, 2a_1, 3a_2) $.  Find an isometry $ S $ such that $ T = S \sqrt{T^* T} $.

\item (Axler 7.29) Let $ T $ be a linear operator an inner product space $ V $.  Prove that $ T $ is invertible if and only if $ 0 $ is not a singular value of $ T $.
\end{enumerate}

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