1.
(a) T
(b) F
(c) F
(d) T
(e) F
(f) T
(g) F
(h) F
(i) F
2.
(a) P=12–√(111−1)andD=(300−1)
(d) P=⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜12–√−12–√016–√16–√−26–√13–√13–√13–√⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟andD=⎛⎝⎜−2000−20004⎞⎠⎟
4. Tz is normal for all z∈C, Tz is self-adjoint if and only if z∈R, and Tz is unitary if and only if |z|=1 .
5. Only the pair of matrices in (d) are unitarily equivalent.
25. 2(ψ−ϕ)
26.
(a) ψ−ϕ2
(b) ψ+ϕ2
27.
(a) x=12–√x′+12–√y′ and y=12–√x′−12–√y′
The new quadratic form is 3(x′)2−(y′)2.
(c) x=313−−√x′+213−−√y′ and y=−213−−√x′+213−−√y′
The new quadratic form is 5(x′)2−8(y′)2.
29.
(c) Q=⎛⎝⎜⎜⎜⎜⎜⎜⎜⎜⎜12–√12–√013–√−13–√13–√−6–√66–√66–√3⎞⎠⎟⎟⎟⎟⎟⎟⎟⎟⎟andR=⎛⎝⎜⎜⎜⎜⎜2–√002–√3–√022–√3–√36–√3⎞⎠⎟⎟⎟⎟⎟
(e) x1=3, x2=−5, x3=4
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