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# Fundamental Mathmatics

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1. a) State the Lagrange Theorem explaining any terms you use.
b) Let alpha, a member of S_11, be the permutation given by

alpha(1) = 7, alpha(2) = 5, alpha(3) = 1, alpha(4) = 2, alpha(5) = 8,
alpha(6) = 9, alpha(7) = 10, alpha(8) = 4, alpha(9) = 11, alpha(10) = 3, alpha(11) = 6.

Decompose the permutation alpha first as a product of disjoint cycles and then as a product of transpsitions. What are the order and sign of alpha and alpha^-1?

c) Show that if H and K are subgroups of a group G, then the intersect of H and K is also a subgroup of G. Show that if H and K have orders 9 and 8, respectively, then the intersect of H and K contains only one element.

2. State the First Isomorphism Theorem explaining any terms you use.
b) Let G = C* be the multiplicative group of nonzero complex numbers. Is the map f: G --> G a homomorphism, provided f is given by i) f(z) = iz, ii) f(z) = z^2, iii) f(z) = |z|, iv) f(z) = z-bar? Justify your answer.
c) Let K be a field and let G be the set of all matrices of the form
a b
0 c
where a, b, c is a member of K and a =/ 0, c =/ 0. Prove that G is a group under matrix multiplication. Prove that the map g: G --> K* x K* defined by
g * the matrix: a b = (a, c)
0 c
is a homomorphism. Here K* is the set of all non-zero elements of K, considered as a multiplicative group. Prove that the kernel of g is isomorphic to the additive group of the filed K. Deduce that the set H consisting of matrices of the form
1 b
0 1
is a normal subgroup of G and the quotient group G/H is isomorphic to K* xx K*. State clearly all results that you used.

3. a) Let V be a vector space over a field K and let f: V --> V be a linear map. Suppose v is a non-zero element of V and lambda is a member of K. Explain what it means to say that v is an eigenvector of f with eigenvalue lambda. Prove that V has a basis consisting of eigenvectors of f if and only if it has a basis with respect to which the matrix representing f is diagonal.
b) Let f: R --> R be a linear map given by
f * binomial (x y) = 1 -2 * (x y)
3 -1

Find the matrix A that corresponds to the mapping f in the basis
u_1 = (1 1), u_2 = (0 1).

c) Find the characteristic and minimal polynomials of the linear map f: R^3 --> R^3 given by the matrix
2 0 0
B = 1 0 1
1 -2 3

Is there a basis for R^3 for which the matrix of f is diagonal? Justify your answer.

4. a) Let V be an inner product spave and let g: V --> V be a map. Explain what it means to say that g is an isometry. Define what it means for a square matrix to be orthogonal. Prove that the product of two orthogonal matrices is orthogonal. Explain the relationship between orthogonal matrices and isometries.

b) Find a, b and c such that the matrix

1/3 0 a
2/3 1/sqrt(2) b
2/3 -1/sqrt(2) c

is orthogonal. Does this condition determine a, b and c uniquely?

c) Let V be a subspace of R defined by

V = {(x_1, x_2, x_3, x_4) is a member of R^4 | x_1 - x_2 + x_3 - x_4 = 0}

Find an orthonormal basis of V.

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## SOLUTION This solution is FREE courtesy of BrainMass!

1. (a) The Lagrange Theorem states that the order of any subgroup H of a finite group G divides the order of G.

(b) We are given the permutation of given by

We wish to write as the product of disjoint cycles and then as a product of transpositions. We also wish to determine the order and sign of and

To find the disjoint cycles in the product representation of , we iterate starting with a given index For we have and Thus we have found the cycle in the product representation of . Next we start with which is the smallest unused index. Now we have and Thus we have found the cycle Next we start with which is the smallest unused index. Now we have and Thus we have found the cycle Since this exhausts all indices, we have following product representation of :

To write as a product of transpositions, we note that the cycle may be written as Thus we have

Since is the product of eight transpositions, which is an even number, we see that is an even transposition, whence its sign is Since the inverse of every permutation has the same signature as the original permutation, we see that also has sign

Now the order of (and , which is necessarily the same) is equal to the least common multiple of the orders of each of the disjoint cycles in its product representation. Thus, the orders of and are both equal to

(c) We wish to show that if H and K are both subgroups of a group G, then is also a subgroup of G. We also wish to show that if H and K have orders 9 and 8 respectively, then has just one element.

To show that is a subgroup of G, it suffices to show that for every pair of elements a and b in , the element of G also lies in . Since H and K are both subgroups of G, and a and b each belong to both H and K, we see that belongs to both H and K, whence it belongs to . Thus we see that is a subgroup of G.

Now since is a group and is a subset of both H and K, it is a subgroup of both H and K. Thus, by Lagrange's Theorem, the order of must divide both the order of H and the order of K, and hence it must also the greatest common divisor of these orders. Now if H has order 9 and K has order 8, then the order of must divide Thus we see that has order 1, whence it has just one element.

2. (a) The First Isomorphism Theorem states that if is a group homomorphism, then the following conditions hold:

(i) The kernel of f, denoted as , is a normal subgroup of G.
(ii) The image of f, denoted as is a subgroup of G.
(iii) The quotient group is isomorphic to .

A group homomorphism is a function from a group G to a group H such that for every pair of elements a and b of G, we have

The kernel of a group homomorphism is the set of all elements g of G such that

The image of a group homomorphism is the set of all elements of H of the form , where g is an element of G.

A subgroup of a group G is a subset of G that is also a group.

A normal subgroup H of a group G is a subgroup is a subgroup of G such that for every element g in G and every element h in H, the element of G also belongs to H.

Let G be a group with normal subgroup H. The quotient group is the set of left cosets of H, where g ranges over all elements of G.

Two groups G and H are isomorphic if there exists a group homomorphism that is both one-to-one and onto.

(b) We are given the multiplicative group of nonzero complex numbers. We wish to determine whether each of the following functions is a group homomorphism.

(i)

This is not a group homomorphism since is not equal to 1.

(ii)

This is a group homomorphism since we have

(iii)

This is a group homomorphism since we have

(iv)

This is a group homomorphism since we have

(c) Let K be a field and let G be the set of all matrices of the form

where a, b, and c belong to K with a and c both nonzero. We wish to show that G is a group under matrix multiplication. We also wish to show that the map given by

is a group homomorphism. We also wish to show that the kernel of g is isomorphic to the additive group of K. Finally, we wish to show that the set H consisting of all matrices of the form

,

where b is in K, is a normal subgroup of G and that the quotient group is isomorphic to .

To see that G is a group under matrix multiplication, we first note that G is associative since matrix multiplication is associative. Next we note that

is the identity element of G, i.e., we have for all elements of G. Finally we note that since

every element of G is invertible. Thus we see that G is a group under matrix multiplication.
To see that g is a group homomorphism, we note that

whence

Now the kernel of g is the set of all matrices

in G such that

Thus we see that the kernel of G is equal to the set H of elements of G of the form

.

Since we have

we see that the kernel of G is isomorphic to the additive group of K.

Finally, by the First Isomorphism Theorem, we see that H is a normal subgroup of G and the quotient group is isomorphic to .
3. (a) Let V be a vector space over a field K and let be a linear map. Suppose v is a nonzero element of V and is an element of K. We wish to explain what it means to say that v is an eigenvector of f with eigenvalue . We also wish to prove that V has a basis consisting of eigenvectors of f if and only if it has a basis with respect to which the matrix representing f is diagonal.

We say that v is an eigenvector of f with eigenvalue if v is nonzero and we have

Suppose that V has a basis consisting of eigenvectors of f. Then we have for every index i from 1 to n. Now the matrix representing f is the matrix A such that for every vector v in V. Thus we have for every index i from 1 to n. Thus, with respect to the basis B, we have

whence A is diagonal with respect to this basis. Conversely, suppose A is diagonal with respect to some basis of V. Then we have , where are the diagonal elements of A. Thus we see that B consists of eigenvectors of f.

(b) We are given the linear map defined by

We wish to find the matrix A corresponding to the mapping f in the basis where

We have

and

Now we have

where and Thus we have

whence

Plugging the first equation into the second, we obtain whence Plugging the third equation into the fourth, we obtain whence Thus we have

(c) We wish to find the characteristic and minimal polynomials of the linear map given by the matrix

We also wish to determine whether there is a basis of for which the matrix of f is diagonal.

The characteristic polynomial of f is given by

Thus the minimal polynomial of f is either or To see whether it is equal to , we compute

Thus we see that Since the minimal polynomial of f is not equal to the characteristic polynomial, we see that f is not diagonalizable.
4. (a) We are given an inner product space V and a map We say g is an isometry if for every pair of vectors u and v in V, we have

A matrix A is orthogonal if

We wish to show that the product of two orthogonal matrices is orthogonal.

Let A and B be orthogonal matrices of the same dimensions. Then we have

whence AB is orthogonal.

The map is an isometry if and only if g is a linear map and the matrix of g is orthogonal.

(b) We wish to find a, b, and c such that the matrix

is orthogonal.

The matrix A is orthogonal if and only if the rows of A form an orthonormal basis of Thus we have

whence and hence We also have

Thus we have whence We also have

Thus we have whence

We also have

whence Thus we see that a and b must have opposite signs, whence and

We also have

whence Thus we see that a and c must also have opposite signs, whence Thus we have the two solutions

(c) We are given the subspace V of defined by

We wish to find an orthonormal basis for V.

First we find a basis for V. It is easy to see that

is a basis for V.

To find an orthonormal basis, we apply the Gram-Schmidt process. First we find an orthogonal basis of V. We have

We also have

Finally we have

Thus we have the following orthogonal basis for V:

Finally, to find an orthonormal basis for V, we need to normalize each of the vectors in . We have

We also have

Finally we have

Thus we have the following orthonormal basis for V:

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