Lv 132 points

# Raveen

Questions12

1. Let u=(1,2,1) , v=(-1,1,1), w =(3,0,-1) in R³

a) Show that w ∈ Ru ⊕ Rv

I've solved a system of linear equations using matrix, the result is a unique solution

w = (1)u -2(v), can I say that w ∈ Ru ⊕ Rv because there`s a unique solution which is 1 and - 2 ?

b) Let f be a linear application in R³, for which values of a∈ R, there exists an endomorphism such that

f(u) = (1,1,0), f(v) = 0,1,1) and f(w)= (1,-1,a).

2. Let M be a ∈ R subspace of dimension 2. Let f be a linear application in M such that f o f = - idM

a) Let x ∈ M\{0}. Show that B = (x,f(x)) is a basis of M.

b) Let u = ax+bf(x) ∈ M, find the coordinates of f(u) in the basis B.

any help will be appreciated !!

how can i prove with epsilon-delta that the limit of :

(-1)ⁿ / (2ⁿ⁺¹) + (-1)ⁿ⁺¹ =0

i know that the absolute value of (-1)ⁿ is 1 but i don`t know how to deal with the absolute value of (2ⁿ⁺¹) + (-1)ⁿ⁺¹.

any help will be appreciated

• ### Real analysis help?

Let {an} be a recursive sequence defined by : a₁ = √2 and an = √(2an-₁) for all n ­>= 2

Show that the sequence is a monotonic sequence and obtain it`s limit.

Let {Xn} be a sequence of real numbers such that Xn -> a. If Xn < = 1 for all n ∈ N.

Prove that a <= 1.

I know that the limit of Xn is a, plus the sequence is bounded above.

i can't find anything to prove that a <= 1 ...

calculate the limit of

2^n + (-1)^n / 2^n+1 + (-1)^n+1

as n approaches infinity

Let E = Mnxn(K) and A = GLn(K) ( set of invertible matrices)

is A a subspace of E ? If not show that span(A) = E .

I know that

A is not a subspace of E because :

A is not closed under scalar multiplication.

A is not closed under addition as well.

But I don't know to prove span(A) = E ...

• ### Prove a function is injective?

Prove that the function f : N -> N is defined by f(n) = 0 + 1 + 2 + ... + n.

I'm not sure of my proof : ( by contrapositive)

for all x1, x2 ∈ N. Assume that f(x1) = f(x2)

f(x1) = 0 + 1 + 2 + ... + x1.

f(x2) = 0 + 1 + 2 + ... + x2

0 + 1 + 2 + ...+ x1 = 0 + 1 + 2 + ... + x2.

hence, x1 = x2.

is it correct ?

1) is the set of all positive real numbers is a vector space over the rationnals ?

I know the answer is no but i'm not sure how to justify it...

if I say R+ is not a vector space will it be enough ? ( because there`s no neutral element and no additive inverse since it`s in R+)

- the multiplication of an element in Q by an element in R+ would NOT be in R+

2) Tell if F is a vector subspace of the vector space E :

F = {f ∈ E ∣ f(-2) = 0} where E = { f : R → R ).

I think yes because if f(-2) = g (-2) = 0, then f(-2) + g(-2) = 0 and c(f(-2))=0

I'm having a hard time with proof...

3) Let E be vector space. Let F, G, H 3 vector subspaces of E.

a) Show that F ∪ G is a vector subspace of E if and only if F ⊆ G or G ⊆ F.

b) Deduct that F ∪ G is a vector subspace of E if and only if F = F ∪ G or G = F ∪ G.

c) Suppose that G ⊆ F. Show that F ∩ (G + H) = G + ( F∩H).

any help will be appreciated !

Let A(matrix 3x3) =

1 5 2

0 1 7

2 11 11

Consider the linear transformation associated to A, TA : R^3 ---> R^3 , TA(X) = AX.

Consider the ordered basis B and C :

B =

2 1 1

1 , 2 , 1

1 1 2

C =

1 0 0

0 , 1 , 0

0 0 1

a) Compute C[TA]C

b) Compute C[TA]B

c) Compute B[TA]B

In the complex plan, consider the regular pentagon whose vertices are root of unity :

1, z= e^(i2pi5), z^2, z^3 = z^(-2), z^4 = z^(-1) = conjugate of z.

Show that all vertices other than 1 are roots of the polynomial

P(X) = X^4 + X^3 + X^2 + X + 1. i.e.

P(X) = (X-z) (X-z^2) (X-z^3) (Z-z^4)

• ### Math proof by induction help please !?

prove by induction :

n n

∑ k^3 = ( ∑ i )^2

k=1 i=1

Consider a cube ABCDEFGH where J,K and L are the respective centers of the faces (ABEF), (EFGH) and (BCGF). F is equidistan to points J,K and L, D is equidistants of points J,K and L. (please refer to the figure)

Prove FD is orthogonal of the plan (JKL).

I can't use vectors to prove this...

The only thing I could think of is the right bissector since F and D are equidistant but the distant from F to the plane is not equal to the distant from D to the plane....

This is new for me to work with 3 dimensions

Any help will be appreciated !