Using definitions of dot product and cross product, show that dot product and cross product are distributive (a) If the three vectors are coplanar. (b) in general.

also,

Is the cross product of two vectors associative i.e. A × (B × C ) = (A × B ) × C ? If so prove it. If not provide a counter example.

Let G be a cyclic group generated by an element a.

a) Prove that if aⁿ = e for some n ∈ Z, then G is finite.

b) Prove that if G is an infinite cyclic group then it contains no nontrivial finite subgroups. (Hint: use part (a))

Problem 3. Let F ⊆ E be a field extension.

(i) Suppose α ∈ E is algebraic of odd degree over F. Prove that F(α) = F(α^2 ). Hints: look at the tower of extensions F ⊆ F(α^2 ) ⊆ F(α) and their degrees.

(ii) Let S be a (possibly infinite) subset of E. Assume that every element of S is algebraic over F. Prove that F(S) = F[S]

diff eq

What, if anything, do the theorems of this chapter allow you to conclude about the existence and uniqueness of solutions to the following initial value problems.

y′′+ty′−t^2y=0; y(0)=0,y′(0)=0

You have just started a new job that offers a retirement savings account. You have two options: You can invest 5% of your monthly wages at 2% OR You can invest 4% of your monthly wages at 4%. Both are compounded monthly. b. Assume that you will always make $45,000 annually, how much will you have saved with the better plan after 15 years? c.Assume that you will always make $45,000 annually, how much will you have saved with the better plan after 25 years?

Exercise 4.8. For each of the following, state whether it is true or false. If

true, prove. If false, provide a counterexample.
(i) LetX beasetfromRn. ThesetX isclosedifandonlyifX isconvex.

(ii) Let X and Y be sets from Rn. If X ∩Y is closed and convex then Xand Y are both closed and convex sets.

(iii) LetX beanopensetandY ⊆X. IfY ≠∅,thenY isaconvexset.

(iv) SupposeX isanopensetandY isaconvexset. IfX∩Y ⊂X then

X∪Y isopen.

Prove If C is a binary self-dual  code, show that every codeword has even weight. Furthermore, prove if each row of the generator matrix of C has weight

divisible by 4, then so does every codeword.

B. Five bowls are labeled 1,2,3,4,5. Bowl i contains i white and 5 − i black ping pong balls, for i = 1,2,3,4,5. A bowl is randomly selected, and 2 ping pong balls are selected from that bowl at random without replacement. Both selected balls were white. What is the probability they were selected from bowl 1? 2? 3? 4? 5?

B. Five bowls are labeled 1,2,3,4,5. Bowl i contains i white and 5 − i black ping pong balls, for i = 1,2,3,4,5. A bowl is randomly selected, and 2 ping pong balls are selected from that bowl at random without replacement. Both selected balls were white. What is the probability they were selected from bowl 1? 2? 3? 4? 5?

Prove that the minimum distance of a linear code is the minimum weight of any nonzero codeword.

Match the following:

Let X = {1,2,3,4}, Classify the relations of X on X
___ {(1,4),(1,2)}
___{ (1,4),(4,1),(2,3) }
___{ (1,4),(4,4),(2,3),(3,3)}
___{ (1,1),(4,4),(2,2),(3,3) }

a. Is a function
b. Is a relation
c. Is transitive
d. Is a relation of equivalence
e. Is not a function

1. Show that 11,111,111 and 3,333,333 are relatively prime using the Extended Euclidean Algorithm.

2. Use the EEA to find the GCD of 6,327 and 10,101.

3. Find the additive inverse of 3,333,333 modulo 11,111,111. Verify.

4. Find the multiplicative inverse of 3,333,333 modulo 11,111,111.Verify.

5. What is the orbit of 3 in the group Z_7 under multiplication modulo 7? Is 3 a generator?

Find either a linear or an exponential function that models the data in the table.

​f(x)=

​(Use integers or fractions for any numbers in the​ expression.)

Note: Please show how to solve using TI-84 Plus calculator