1.

1. (a) T

2. (b) F

3. (c) F

4. (d) T

5. (e) T

6. (f) T

7. (g) F

8. (h) T

2.

1. (a) T is self-adjoint. An orthonormal basis of eigenvectors is $\left\{{\displaystyle \frac{1}{\sqrt{5}}}(1,-2),{\displaystyle \frac{1}{\sqrt{5}}}(2,1)\right\},$ with corresponding eigenvalues 6 and 1.

2. (c) T is normal, but not self-adjoint. An orthonormal basis of eigenvectors is $\left\{{\displaystyle \frac{1}{2}}(1+i,\sqrt{2}),{\displaystyle \frac{1}{2}}(1+i,-\sqrt{2})\right\}$ with corresponding eigenvalues $2+{\displaystyle \frac{1+i}{\sqrt{2}}}$ and $2-{\displaystyle \frac{1+i}{\sqrt{2}}}.$

3. (e) T is self-adjoint. An orthonormal basis of eigenvectors is $\{{\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{rr}0& 1\\ 1& 0\end{array}\right),{\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{rr}1& 0\\ 0& 1\end{array}\right),{\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{rr}0& -1\\ 1& 0\end{array}\right),{\displaystyle \frac{1}{\sqrt{2}}}\left(\begin{array}{rr}-1& 0\\ 0& 1\end{array}\right)\}$ with corresponding eigenvalues 1, 1,$-1$, 1.