Exercise 1 Without computing the eigenvalues and eigenvectors, explain why the ma trices -1 1-2 0 K- 1+2 0 → 0 i 1 are diagonalizable over C and why is diagonalizable over R -21 112 12 2 Exercise 2 Consider the vector space V - (JC (je.4.C)| (a) (b)) at the inmer product (1.3) - de Heigle). (a) Show that the operator (Df)(r)=df/dr is skew-adjoint, messing that D¹-D. (b) Show that the operator (D³) df/d³ is self-adjoist, messing that (D)-D Exercise 3 On the vector space V=C(0, 1) of real continous functions define (1.9) - Lde frigte).z (a) Show that (.) defines an inter product (you may use the fact that (f.9)e fde f(a) giz) is an inner product) (b) Let - be the oem induced by the scalar product (3). Calculate (e) Show the inequality (4)
Exercise 1 Without computing the eigenvalues and eigenvectors, explain why the ma trices -1 1-2 0 K- 1+2 0 → 0 i 1 are diagonalizable over C and why is diagonalizable over R -21 112 12 2 Exercise 2 Consider the vector space V - (JC (je.4.C)| (a) (b)) at the inmer product (1.3) - de Heigle). (a) Show that the operator (Df)(r)=df/dr is skew-adjoint, messing that D¹-D. (b) Show that the operator (D³) df/d³ is self-adjoist, messing that (D)-D Exercise 3 On the vector space V=C(0, 1) of real continous functions define (1.9) - Lde frigte).z (a) Show that (.) defines an inter product (you may use the fact that (f.9)e fde f(a) giz) is an inner product) (b) Let - be the oem induced by the scalar product (3). Calculate (e) Show the inequality (4)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![Exercise 1
Without computing the eigenvalues and eigenvectors, explain why the ma
trices
-1 1-2 0
K- 1+2 0 7 G-
0
i 1
are diagonalizable over and why
-21
112
-3 2-1
12 2
is diagonalizable over R
Exercise 2
Consider the vector space V- (C(C) | F(e)-f()) the
inmer product
- Lde Fergle).
(a) Show that the operator (Dfr)=df/dr is skew-adjoint, messing
that D¹-D.
(b) Show that the operator (DJ)(e) f/² is self-adjoit, messing
Exercise 3
On the vector space V=C(10.13) of real continous functions define
(1.9) - [ar f(xg(0)z
R
(a) Show that (.) defines an inter product (you may use the fact that
(7.9)edz fa)(a) is an inner product)
(b) Let I be the orm induced by the scalar product (3). Calculate
and 1/(1+²)
(e) Show the inequality
2
(4)
Exercise 4
On the vector space V=C(-1,1]) define the inner product
(1.9) - - Lde f(x) g(2²
(5)
Apply the Gramm-Schmidt process to the vectors 1 and 2 to obtain an or
thonormal basis for the subspace spanned by the two vectors](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbf9c0aa8-f0c2-40a6-a625-b5a0e0c97da9%2Fdac868e8-3cad-43f4-85c1-f9c061a8607b%2Fnr7134c_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 1
Without computing the eigenvalues and eigenvectors, explain why the ma
trices
-1 1-2 0
K- 1+2 0 7 G-
0
i 1
are diagonalizable over and why
-21
112
-3 2-1
12 2
is diagonalizable over R
Exercise 2
Consider the vector space V- (C(C) | F(e)-f()) the
inmer product
- Lde Fergle).
(a) Show that the operator (Dfr)=df/dr is skew-adjoint, messing
that D¹-D.
(b) Show that the operator (DJ)(e) f/² is self-adjoit, messing
Exercise 3
On the vector space V=C(10.13) of real continous functions define
(1.9) - [ar f(xg(0)z
R
(a) Show that (.) defines an inter product (you may use the fact that
(7.9)edz fa)(a) is an inner product)
(b) Let I be the orm induced by the scalar product (3). Calculate
and 1/(1+²)
(e) Show the inequality
2
(4)
Exercise 4
On the vector space V=C(-1,1]) define the inner product
(1.9) - - Lde f(x) g(2²
(5)
Apply the Gramm-Schmidt process to the vectors 1 and 2 to obtain an or
thonormal basis for the subspace spanned by the two vectors
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