linear-guest

14.7 Review Problems 275
7. (a) Show that if Q is an orthogonal n × n matrix, then
u v = (Qu) (Qv) ,
for any u, v ∈ R n . That is, Q preserves the inner product.
(b) Does Q preserve the outer product?
(c) If the set of vectors {u 1 , . . . , u n } is orthonormal and {λ 1 , · · · , λ n }
is a set of numbers, then what are the eigenvalues and eigenvectors
of the matrix M = ∑ n
i=1 λ iu i u T i ?
(d) How would the eigenvectors and eigenvalues of this matrix change
if we replaced {u 1 , . . . , u n } by {Qu 1 , . . . , Qu n }?
8. Carefully write out the Gram-Schmidt procedure for the set of vectors
⎧⎛
⎞ ⎛ ⎞ ⎛ ⎞⎫
⎨ 1 1 1 ⎬
⎝1⎠ , ⎝−1⎠ , ⎝ 1⎠
⎩
⎭ .
1 1 −1
Is it possible to rescale the second vector obtained in the procedure to
a vector with integer components?
9. (a) Suppose u and v are linearly independent. Show that u and v ⊥
are also linearly independent. Explain why {u, v ⊥ } is a basis for
span{u, v}.
Hint
(b) Repeat the previous problem, but with three independent vectors
u, v, w where v ⊥ and w ⊥ are as defined by the Gram-Schmidt
procedure.
10. Find the QR factorization of
⎛
1 0
⎞
2
M = ⎝−1 2 0⎠ .
−1 −2 2
11. Given any three vectors u, v, w, when do v ⊥ or w ⊥ of the Gram–Schmidt
procedure vanish?
275