MFDS Homework 1 Solved

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2.1          We consider (R\{−1},?), where

a ? b := ab + a + b,               a,b R\{−1}                                (2.134)

  1. Show that (R\{−1},?) is an Abelian group.
  2. Solve

3 ? x ? x = 15

in the Abelian group (R\{−1},?), where ? is defined in (2.134).

2.2      Let n be in N\{0}. Let k,x be in Z. We define the congruence class k¯ of the integer k as the set

.

We now define Z/nZ (sometimes written Zn) as the set of all congruence classes modulo n. Euclidean division implies that this set is a finite set containing n elements:

Zn = {0,1,…,n − 1}

For all a,b ∈ Zn, we define

a b := a + b

  1. Show that (Zn,⊕) is a group. Is it Abelian?
  2. We now define another operation ⊗ for all a and b in Zn as

a b = a × b,                                                   (2.135)

where a × b represents the usual multiplication in Z.

Let n = 5. Draw the times table of the elements of Z5\{0} under ⊗, i.e., calculate the products a b for all a and b in Z5\{0}.

Hence, show that Z5\{0} is closed under ⊗ and possesses a neutral

element for ⊗. Display the inverse of all elements in Z5\{0} under ⊗. Conclude that (Z5\{0},⊗) is an Abelian group.

  1. Show that (Z8\{0},⊗) is not a group.
  2. We recall that the B´ezout theorem states that two integers a and b are relatively prime (i.e., gcd(a,b) = 1) if and only if there exist two integers

u and v such that au + bv = 1. Show that (Zn\{0},⊗) is a group if and only if n ∈ N\{0} is prime.

  • Consider the set G of 3 × 3 matrices defined as follows:

We define · as the standard matrix multiplication.

Is (G,·) a group? If yes, is it Abelian? Justify your answer.

  • Compute the following matrix products, if possible:

a.

1     21     1     0

4     50     1     1

7     8       1     0     1

b.

c.

1     1     01     2     3

0     1     14     5     6

1     0     1       7     8     9

d.

e.

2.5 Find the set S of all solutions in x of the following inhomogeneous linear systems Ax = b, where A and b are defined as follows:

a.

A , b

b.

A , b

2.6 Using Gaussian elimination, find all solutions of the inhomogeneous equation system Ax = b with

A , b .

  • Find all solutions in x of the equation system Ax = 12x,

where

A

and P3i=1 xi = 1.

  • Determine the inverses of the following matrices if possible:

a.

A

b.

1     0     1     0

A = 0     1     1    0

1     1     0     1

1     1     1     0

2.9          Which of the following sets are subspaces of R3?

  1. A = {(λ,λ + µ3µ3) | λ,µ R}
  2. B = {(λ2,λ2,0) | λ R}
  3. Let γ be in R.

C = {(ξ123) ∈ R3 | ξ1 − 2ξ2 + 3ξ3 = γ}

  1. D = {(ξ123) ∈ R3 | ξ2 Z}

2.10 Are the following sets of vectors linearly independent?

a.

x , x , x

b.    
1

2

 

x1 = 1 ,

 

0

0

1

1

 

x2 = 0 ,

 

1

1

1

0

  x3 = 0

 

1

1

2.11 Write

y

as linear combination of

x , x , x

2.12 Consider two subspaces of R4:

−1  2  −3

U1 = span[                                        = span[ 2 ,−2 , 6 ].

2   0  −2

1             0             −1

Determine a basis of U1 U2.

2.13 Consider two subspaces U1 and U2, where U1 is the solution space of the homogeneous equation system A1x = 0 and U2 is the solution space of the homogeneous equation system A2x = 0 with

1       0         1                      3     −3      0

A1 = 1      −2       −1 , A2 = 1         2      3 .

2       1         3                      7     −5      2

1       0        1                             3     −1      2

  1. Determine the dimension of U1,U2.
  2. Determine bases of U1 and U2.
  3. Determine a basis of U1 U2.
    • Consider two subspaces U1 and U2, where U1 is spanned by the columns of A1 and U2 is spanned by the columns of A2 with
                            1       0         1                      3

A1 = 1      −2        −1, A2 = 1

2       1         3                      7

1       0        1                             3

a.    Determine the dimension of U1,U2

b.   Determine bases of U1 and U2

c.    Determine a basis of U1 U2

−3

2

−5

−1

0

3

2.

2

  • Let F = {(x,y,z) ∈ R3 | x+yz = 0} and G = {(ab,a+b,a−3b) | a,b R}.
  1. Show that F and G are subspaces of R3.
  2. Calculate F G without resorting to any basis vector.
  3. Find one basis for F and one for G, calculate FG using the basis vectors previously found and check your result with the previous question.

2.16 Are the following mappings linear?

  1. Let a,b ∈ R.

Φ : L1([a,b]) → R

where L1([a,b]) denotes the set of integrable functions on [a,b].

b.

Φ : C1 C0

f 7→ Φ(f) = f0 ,

where for k > 1, Ck denotes the set of k times continuously differentiable functions, and C0 denotes the set of continuous functions.

c.

Φ : R → R

x 7→ Φ(x) = cos(x)

d.

Φ : R3 → R2

xx

  1. Let θ be in [0,2π[ and

Φ : R2 → R2

xx

2.17 Consider the linear mapping

Φ : R3 → R4

x1                 3x1 + 2x2 + x3  x1 + x2 + x3 

Φx2 =    x1 − 3x2     x3

2x1 + 3x2 + x3

Find the transformation matrix AΦ.

Determine rk(AΦ).

Compute the kernel and image of Φ. What are dim(ker(Φ)) and dim(Im(Φ))?

2.18 Let E be a vector space. Let f and g be two automorphisms on E such that f g = idE (i.e., f g is the identity mapping idE). Show that ker(f) = ker(g f), Im(g) = Im(g f) and that ker(f) ∩ Im(g) = {0E}.

2.19 Consider an endomorphism Φ : R3 → R3 whose transformation matrix (with respect to the standard basis in R3) is

A .

  1. Determine ker(Φ) and Im(Φ).
  2. Determine the transformation matrix A˜ Φ with respect to the basis

,

i.e., perform a basis change toward the new basis B.

2.20 Let us consider b vectors of R2 expressed in the standard basis of R2 as

b

and let us define two ordered bases B = (b1,b2) and of R2.

  1. Show that B and B0 are two bases of R2 and draw those basis vectors.
  2. Compute the matrix P1 that performs a basis change from B0 to B.
  3. We consider c1,c2,c3, three vectors of R3 defined in the standard basis

of R3 as

c

and we define C = (c1,c2,c3).

  • Show that C is a basis of R3, e.g., by using determinants (see Section 4.1).
  • Let us call the standard basis of R3. Determine the matrix P2 that performs the basis change from C to C0.
  1. We consider a homomorphism Φ : R2 −→ R3, such that

Φ(b1 + b2)           = c2 + c3

Φ(b1 b2)       =         2c1 c2 + 3c3

where B = (b1,b2) and C = (c1,c2,c3) are ordered bases of R2 and R3, respectively.

Determine the transformation matrix AΦ of Φ with respect to the ordered bases B and C.

  1. Determine A0, the transformation matrix of Φ with respect to the bases

B0 and C0.

  1. Let us consider the vector x ∈ R2 whose coordinates in B0 are [2,3]>.

In other words, x.

  • Calculate the coordinates of x in B.
  • Based on that, compute the coordinates of Φ(x) expressed in C.
  • Then, write Φ(x) in terms of .
  • Use the representation of x in B0 and the matrix A0 to find this result directly.

 

96                                                                                                                             Analytic Geometry

Exercises

3.1                                  Show that h·,·i defined for all x = [x1,x2]> ∈ R2 and y = [y1,y2]> ∈ R2 by

hx,yi := x1y1 − (x1y2 + x2y1) + 2(x2y2)

is an inner product.

3.2              Consider R2 with h·,·i defined for all x and y in R2 as

 .

=:A

Is h·,·i an inner product? 3.3      Compute the distance between

x , y

using

  1. hx,yi := x>y
  2. hx,yi := x>Ay , A := 4 Compute the angle between

x , y

using

  1. hx,yi := x>y
  2. hx,yi := x>By , B :=

3.5                      Consider the Euclidean vector space R5 with the dot product. A subspace

U ⊆ R5 and x ∈ R5 are given by

−1

−9

U = span[, x = −1.

      

4

1

  1. Determine the orthogonal projection πU(x) of x onto U
  2. Determine the distance d(x,U)

3.6          Consider R3 with the inner product

 .

Furthermore, we define e1,e2,e3 as the standard/canonical basis in R3.