# MFDS Homework 1 Solved

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2.1Â Â Â Â Â Â Â Â Â  We consider (R\{âˆ’1},?), where

a ? b := ab + a + b,Â Â Â Â Â Â Â Â Â Â Â Â Â Â  a,b âˆˆ R\{âˆ’1}Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  (2.134)

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

3 ? x ? x = 15

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

2.2Â Â Â Â Â  Let n be in N\{0}. Let k,x be in Z. We define the congruence class kÂ¯ 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 âˆ’ 1}

For all a,b âˆˆ Zn, we define

a âŠ• b := a + b

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

a âŠ— b = a Ã— b,Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  (2.135)

where a Ã— b represents the usual multiplication in Z.

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

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

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

1. Show that (Z8\{0},âŠ—) is not a group.
2. We recall that the BÂ´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},âŠ—) is a group if and only if n âˆˆ N\{0} is prime.

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

We define Â· as the standard matrix multiplication.

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

• Compute the following matrix products, if possible:

a.

ï£®1Â Â Â Â  2ï£¹ï£®1Â Â Â Â  1Â Â Â Â  0ï£¹

ï£°4Â Â Â Â  5ï£»ï£°0Â Â Â Â  1Â Â Â Â  1ï£»

7Â Â Â Â  8Â Â Â Â Â Â  1Â Â Â Â  0Â Â Â Â  1

b.

c.

ï£®1Â Â Â Â  1Â Â Â Â  0ï£¹ï£®1Â Â Â Â  2Â Â Â Â  3ï£¹

ï£°0Â Â Â Â  1Â Â Â Â  1ï£»ï£°4Â Â Â Â  5Â Â Â Â  6ï£»

1Â Â Â Â  0Â Â Â Â  1Â Â Â Â Â Â  7Â Â Â Â  8Â Â Â Â  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.

ï£®1Â Â Â Â  0Â Â Â Â  1Â Â Â Â  0ï£¹

A = ï£¯ï£¯0Â Â Â Â  1Â Â Â Â  1Â Â Â  0ï£ºï£º

ï£°1Â Â Â Â  1Â Â Â Â  0Â Â Â Â  1ï£»

1Â Â Â Â  1Â Â Â Â  1Â Â Â Â  0

2.9Â Â Â Â Â Â Â Â Â  Which of the following sets are subspaces of R3?

1. A = {(Î»,Î» + Âµ3,Î» âˆ’ Âµ3) | Î»,Âµ âˆˆ R}
2. B = {(Î»2,âˆ’Î»2,0) | Î» âˆˆ R}
3. Let Î³ be in R.

C = {(Î¾1,Î¾2,Î¾3) âˆˆ R3 | Î¾1 âˆ’ 2Î¾2 + 3Î¾3 = Î³}

1. D = {(Î¾1,Î¾2,Î¾3) âˆˆ R3 | Î¾2 âˆˆ Z}

2.10 Are the following sets of vectors linearly independent?

a.

x , x , x

 b. ï£®1ï£¹ 2 ï£¯ ï£º x1 = ï£¯ï£¯1ï£ºï£º , ï£¯ ï£º ï£°0ï£» 0 ï£®1ï£¹ 1 ï£¯ ï£º x2 = ï£¯ï£¯0ï£ºï£º , ï£¯ ï£º ï£°1ï£» 1 ï£®1ï£¹ 0 ï£¯ ï£º x3 = ï£¯ï£¯0ï£ºï£º ï£¯ ï£º ï£°1ï£» 1

2.11 Write

y

as linear combination of

x , x , x

2.12 Consider two subspaces of R4:

ï£®âˆ’1ï£¹ ï£® 2 ï£¹ ï£®âˆ’3ï£¹

âˆ’

U1 = span[Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  = span[ï£¯ï£¯ 2ï£ºï£º ,ï£¯ï£¯âˆ’2ï£ºï£º ,ï£¯ï£¯ 6 ï£ºï£º].

ï£° 2 ï£» ï£° 0 ï£» ï£°âˆ’2ï£»

1Â Â Â Â Â Â Â Â Â Â Â Â  0Â Â Â Â Â Â Â Â Â Â Â Â  âˆ’1

Determine a basis of U1 âˆ© 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

ï£®1Â Â Â Â Â Â  0Â Â Â Â Â Â Â Â  1 ï£¹Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  ï£®3Â Â Â Â  âˆ’3Â Â Â Â Â  0ï£¹

A1 = ï£¯ï£¯1Â Â Â Â Â  âˆ’2Â Â Â Â Â Â  âˆ’1ï£ºï£º , A2 = ï£¯ï£¯1Â Â Â Â Â Â Â Â  2Â Â Â Â Â  3ï£ºï£º .

ï£°2Â Â Â Â Â Â  1Â Â Â Â Â Â Â Â  3 ï£»Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  ï£°7Â Â Â Â  âˆ’5Â Â Â Â Â  2ï£»

1Â Â Â Â Â Â  0Â Â Â Â Â Â Â  1Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  3Â Â Â Â  âˆ’1Â Â Â Â Â  2

1. Determine the dimension of U1,U2.
2. Determine bases of U1 and U2.
3. Determine a basis of U1 âˆ© 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
 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  ï£®1Â Â Â Â Â Â  0Â Â Â Â Â Â Â Â  1 ï£¹Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  ï£®3 A1 = ï£¯ï£¯1Â Â Â Â Â  âˆ’2Â Â Â Â Â Â Â  âˆ’1ï£ºï£º , A2 = ï£¯ï£¯1 ï£°2Â Â Â Â Â Â  1Â Â Â Â Â Â Â Â  3 ï£»Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  ï£°7 1Â Â Â Â Â Â  0Â Â Â Â Â Â Â  1Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  3 a.Â Â Â  Determine the dimension of U1,U2 b.Â Â  Determine bases of U1 and U2 c.Â Â Â  Determine a basis of U1 âˆ© U2 âˆ’3 2 âˆ’5 âˆ’1 0ï£¹ 3ï£º 2ï£ºï£» . 2
• Let F = {(x,y,z) âˆˆ R3 | x+yâˆ’z = 0} and G = {(aâˆ’b,a+b,aâˆ’3b) | a,b âˆˆ R}.
1. Show that F and G are subspaces of R3.
2. Calculate F âˆ© G without resorting to any basis vector.
3. Find one basis for F and one for G, calculate Fâˆ©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 âˆˆ R.

Î¦ : L1([a,b]) â†’ R

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

b.

Î¦ : C1 â†’ C0

f 7â†’ Î¦(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.

Î¦ : R â†’ R

x 7â†’ Î¦(x) = cos(x)

d.

Î¦ : R3 â†’ R2

xx

1. Let Î¸ be in [0,2Ï€[ and

Î¦ : R2 â†’ R2

xx

2.17 Consider the linear mapping

Î¦ : R3 â†’ R4

ï£«ï£®x1ï£¹ï£¶Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  ï£®3x1 + 2x2 + x3ï£¹ ï£¯ x1 + x2 + x3 ï£º

Î¦ï£­ï£°x2ï£»ï£¸ = ï£¯ï£°Â Â  x1 âˆ’ 3x2Â Â Â  ï£ºï£» x3

2x1 + 3x2 + x3

Find the transformation matrix AÎ¦.

Determine rk(AÎ¦).

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

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

2.19 Consider an endomorphism Î¦ : R3 â†’ R3 whose transformation matrix (with respect to the standard basis in R3) is

A .

1. Determine ker(Î¦) and Im(Î¦).
2. Determine the transformation matrix AËœ Î¦ 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 Î¦ : R2 âˆ’â†’ R3, such that

Î¦(b1 + b2)Â Â Â Â Â Â Â Â Â Â  = c2 + c3

Î¦(b1 âˆ’ b2)Â Â Â Â Â Â  =Â Â Â Â Â Â Â Â  2c1 âˆ’ c2 + 3c3

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

Determine the transformation matrix AÎ¦ of Î¦ with respect to the ordered bases B and C.

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

B0 and C0.

1. Let us consider the vector x âˆˆ 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 Î¦(x) expressed in C.
• Then, write Î¦(x) in terms of .
• Use the representation of x in B0 and the matrix A0 to find this result directly.

96Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  Analytic Geometry

# Exercises

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

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

is an inner product.

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

Â .

=:A

Is hÂ·,Â·i an inner product? 3.3Â Â Â Â Â  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 :=

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

U âŠ† R5 and x âˆˆ R5 are given by

ï£®âˆ’1ï£¹

ï£¯âˆ’9ï£º

U = span[, x = ï£¯ï£¯âˆ’1ï£ºï£º .

ï£¯Â Â Â Â Â  ï£º

ï£° 4 ï£»

1

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

3.6Â Â Â Â Â Â Â Â Â  Consider R3 with the inner product

Â .

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

• homework1-ie1d5v.zip