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 Bezout 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 21 1 0

4 50 1 1

7 8 1 0 1

b.

c.

1 1 01 2 3

0 1 14 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 = {(, + ³, ³) | , ^R}
2. B = {(²,²,0) | ^R}
3. Let be in R.

C = {(₁,₂,₃) R³ | ₁ 2₂ + 3₃ = }

1. D = {(₁,₂,₃) R³ | ₂ ^Z}

2.10 Are the following sets of vectors linearly independent?

a.

x , x , x

b.
12 x1 = 1 , 00	11 x2 = 0 , 11	10 x3 = 0 11

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 3A1 = 1 2 1 , A2 = 12 1 3 71 0 1 3a. Determine the dimension of U1,U2b. Determine bases of U1 and U2c. Determine a basis of U1 U2

3251

032 .2

• Let F = {(x,y,z) R³ | x+yz = 0} and G = {(ab,a+b,a3b) | 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 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.

: L¹([a,b]) R

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

b.

: C¹ C⁰

f 7 (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.

: R R

x 7 (x) = cos(x)

d.

: R³ R²

xx

1. Let be in [0,2[ and

: R² R²

xx

2.17 Consider the linear mapping

: R³ R⁴

x₁ 3x1 + 2x2 + x3 x1 + x2 + x3

x2 = x1 3x2 x3

2x₁ + 3x₂ + x₃

Find the transformation matrix A.

Determine rk(A).

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

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 : R³ R³ whose transformation matrix (with respect to the standard basis in R³) is

A .

1. Determine ker() and Im().
2. Determine the transformation matrix A 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 : R² R³, such that

(b₁ + b₂) = c₂ + c₃

(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 of with respect to the ordered bases B and C.

1. Determine _A⁰, the transformation matrix of 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 (x) expressed in C.
• Then, write (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 _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.