Linear algebra: please solve q21 and 22 handwritten don't send me the typed solution In Exercises 1-18, s = 1+2i, u = 3 - 2i, v = 4+i,w = 2-i, and z = 1+i. In each exercise, perform theindicated calculation and express the result in the forma + ib.1. ū4. Z + w7. vv10. z²w13. u/v16. (w + v)/u2. Z5. u + ū8. uv11. uw²14. v/u²17. w + izFind the eigenvalues and the eigenvectors for the matri-ces in Exercises 19-24. (For the matrix in Exercise 24,one eigenvalue is λ = 1 + 5i.)19.6 824[49] 20. [44]-1 227. x =In Exercises 27-30, calculate ||x||.1+i23. u + v6. s-s9. s² - w12. s(u² + v)15. s/z18. s - iw1-2i[B]3+i29. x =28. x =3+i2-i30. x =2i[4]321.[34]23.1 -4 -13 2 31 1 322.24.5-5-5-1 4 23 -5 -31-5005 10 01-20 000 2 1In Exercises 25 and 26, solve the linear system.25. (1+i)x+iy = 5 + 4i(1 - i)x - 4y = -11+5i26. (1 - i)x (3+i)y=-5 - i(2+i)x + (1+2i)y= 1+6i4.7 Similarity Transformations and Diagonalization325Suppose that A is an (m × n) matrix and B is an(nx p) matrix. Use Exercise 36 and the propertiesof the transpose operation to give a quick proof that(AB)* = B* A*.38. An (n xn) matrix A is called Hermitian if A* = A.a) Prove that a Hermitian matrix A has only realeigenvalues. [Hint: Observing that x¹ x = x*x,modify the proof of Theorem 17.]

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Answered: In Exercises 1-18, s = 1 + 2i, u = 3 –…

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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Linear algebra: please solve q21 and 22 handwritten don't send me the typed solution

In Exercises 1-18, s = 1+2i, u = 3 - 2i, v = 4+i,
w = 2-i, and z = 1+i. In each exercise, perform the
indicated calculation and express the result in the form
a + ib.
1. ū
4. Z + w
7. vv
10. z²w
13. u/v
16. (w + v)/u
2. Z
5. u + ū
8. uv
11. uw²
14. v/u²
17. w + iz
Find the eigenvalues and the eigenvectors for the matri-
ces in Exercises 19-24. (For the matrix in Exercise 24,
one eigenvalue is λ = 1 + 5i.)
19.
6 8
24
[49] 20. [44]
-1 2
27. x =
In Exercises 27-30, calculate ||x||.
1+i
2
3. u + v
6. s-s
9. s² - w
12. s(u² + v)
15. s/z
18. s - iw
1-2i
[B]
3+i
29. x =
28. x =
3+i
2-i
30. x =
2i
[4]
3
21.
[34]
23.
1 -4 -1
3 2 3
1 1 3
22.
24.
5-5-5
-1 4 2
3 -5 -3
1-5
0
0
5 1
0 0
1-2
0 0
00 2 1
In Exercises 25 and 26, solve the linear system.
25. (1+i)x+iy = 5 + 4i
(1 - i)x - 4y = -11+5i
26. (1 - i)x (3+i)y=-5 - i
(2+i)x + (1+2i)y= 1+6i
4.7 Similarity Transformations and Diagonalization
325
Suppose that A is an (m × n) matrix and B is an
(nx p) matrix. Use Exercise 36 and the properties
of the transpose operation to give a quick proof that
(AB)* = B* A*.
38. An (n xn) matrix A is called Hermitian if A* = A.
a) Prove that a Hermitian matrix A has only real
eigenvalues. [Hint: Observing that x¹ x = x*x,
modify the proof of Theorem 17.]

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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