Description

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 Z_n) as the set of all congruence classes modulo n. Euclidean division implies that this set is a finite set containing n elements:

^Z_n = {0,1,…,n − 1}

For all a,b ∈ Z_n, we define

a ⊕ b := a + b

1. Show that (Z_n,⊕) is a group. Is it Abelian?
2. We now define another operation ⊗ for all a and b in Z_n 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 (Z_n\{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 + µ³,λ − µ³) | λ,µ ∈ ^R}
2. B = {(λ²,−λ²,0) | λ ∈ ^R}
3. Let γ be in R.

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

1. D = {(ξ₁,ξ₂,ξ₃) ∈ 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

2. Determine the dimension of U₁,U₂.
3. Determine bases of U₁ and U₂.
4. 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+y−z = 0} and G = {(a−b,a+b,a−3b) | a,b ∈ ^R}.

3. Show that F and G are subspaces of R³.
4. Calculate F ∩ G without resorting to any basis vector.
5. Find one basis for F and one for G, calculate F∩G 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, C^k 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 A_Φ.

Determine rk(A_Φ).

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 = id_E (i.e., f ◦ g is the identity mapping id_E). Show that ker(f) = ker(g ◦ f), Im(g) = Im(g ◦ f) and that ker(f) ∩ Im(g) = {0_E}.

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 A_Φ 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 π_U(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.