1. Let V be real vector space (possibly infinite-dimensional), S, T ∈ L(V ), and S be in- vertible. Prove λ ∈ C is an eigenvalue of T if and only if λ is an eigenvalue of STS−1. Give a description of the set of eigenvectors of STS−1 associated to an eigenvalue λ in terms of the eigenvectors of T associated to λ.

2. Show that there exist square matrices A, B that have the same eigenvalues, but aren’t similar. Hint: Use the identity matrix as one of the matrices.

Take one step of Newton’s method to approximate a solution to the complex equation z ^5 − 1 = 0, with z0 = i. Simplify your result to identify the real and imaginary parts of your approximation. What is the nearest actual root?

If R is the 2×2 matrices over the real, show that R has nontrivial left and right ideals.?

hello
could you please solve this problem with the clear hands writing to read it please? Also the good explanation to understand the solution is by step by step please

thank

the subject is Modern Algebra

The set D = {b,c, d,l,n,p, s, w} consists of the eight dogs Bingley, Cardie, Duncan, Lily, Nico, Pushkin, Scout, and Whistle. There are three subsets B={d,l,n,s}, F = {c,l,p,s}. and R = {c,n,s,w} of black dogs, female dogs, and retrievers respectively.
(a) Suppose x is one of the dogs in D. Indicate how you can determine which dog x is by asking three yes-or-no questions about x.
(b) Define six subsets of the naturals {1,. .., 64}, each containing 32 numbers, such that you can determine any number n from the answers to the six membership questions for these sets.

(c) (harder) Is it possible to solve part (a) using three sets of dogs that do not have four elements each? What about part (b), with six sets that do not have 32 elements each? Explain your answer.

Reduce the following modular arithmetic WITHOUT THE USE OF A CALCULATOR: PLEASE STATE THE THEOREMS/RULES YOU USE AND EXPLAIN HOW. Thanks!!

a) 104^5 mod 2669

b) 11^132 mod 133

c) 2208^5 mod 2669

d) 7^1000 mod 5

e) 2^247 mod 35

Find all triples of a, b, c satisfying (a, b, c) = 10 and [a, b, c] = 100 simultaneouly

1. Let α < β be real numbers and N ∈ N.

(a). Show that if β − α > N then there are at least N distinct integers strictly between β and α.
(b). Show that if β > α are real numbers then there is a rational number q ∈ Q such β > q > α.

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2. Let x, y, z be real numbers.The absolute value of x is defined by

|x|= x, if x ≥ 0 ,

−x, if x < 0

Define the following statements. Assume that the product of two positive real numbers is positive.

(a). Show that |x−y| = |y−x|.

(b). Show that if z<0 and x ≤ y, then zy ≤ zx.

(c). Show that |x+y| ≤ |x|+|y|.

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Answer both questions plz. Will rate.

Problem 4. From order two to order three:
(a) Find the general solution of y′′′ + 3y′′ + 3y′ + y = 0.
(b) Write a differential equation given that the fundamental system of solutions is

ex,exsinx, excosx.
(c) Compute the general solution of xy′′′ + y′′ = x2. [Answer: C1x ln x + C2x + C3 + x4 .]

prove that the Luxemberg norm is a norm on L_phi

in a cartesian coordinate space, a curved path is defined as y=sin(x).Find the vector that is normal to the path everywhere.(xy:no unit)

Suppose f is a function defined for all real numbers which has a maximum value of 5 and a minimum value of −7. Label each of the following as MUST, MIGHT, or NEVER true. Explain, and if you say might, give an example of yes and an example of no.

A The maximum value of f(|x|) is 7.

B The maximum value of f(|x|) is 5.

C The maximum value of f(|x|) is 0.

D The minimum value of f(|x|) is 7.

E The minimum value of f(|x|) is 5.

F The minimum value of f(|x|) is 0.

G The maximum value of |f(x)| is 7.

H The maximum value of |f(x)| is 5.

I The maximum value of |f(x)| is 0.

J The minimum value of |f(x)| is 7.

K The minimum value of |f(x)| is 5.

L The minimum value of |f(x)| is 0.

Now suppose f is a continuous function defined for all real numbers which has a maximum value of 5 and a minimum value of −7. Which of the answers above stay the same (why?), and which change (to what, and why?)

Find five positive natural numbers u, v, w, x, y such that there is no subset with a sum divisible by 5

We say a graph is k-regular if every vertex has degree exactly k. In each of the following either give a presentation of the graph or show that it does not exist. 1) 3-regular graph on 2018 vertices. 2) 3-regular graph on 2019 vertices.

Consider a function f(x) which satisfies the following properties:

1. f(x+y)=f(x) * f(y)

2. f(0) does not equal to 0

3. f'(0)=1

Then:

a) Show that f(0)=1. (Hint: use the fact that 0+0=0)

b) Show that f(x) does not equal to 0 for all x. (Hint: use y= -x with conditions (1) and (2) above.)

c) Use the definition of the derivative to show that f'(x)=f(x) for all real numbers x

d) let g(x) satisfy properties (1)-(3) above and let k(x) =f(x)/g(x). Show that k(x) is defined for all x and find k'(x). Use this to discover the relationship between f(x) and g(x)

e) Can you think of a function that satisfies (1)-(3)? Could there be more than one such function? Explain why or why not.

f) What do you think would happen if you changed condition (3) so that f'(0)=a for some a>0, rather than 1? Would you still find a familiar function?