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MFDS - homework1 - Solved
Exercises
2.1 We consider (R\{−1},?), where
a ? b := ab + a + b, a,b ∈ R\{−1} (2.134)
a. Show that (R\{−1},?) is an Abelian group.
b. 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
a. Show that (Zn,⊕) is a group. Is it Abelian?
b. 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.
c. Show that (Z8\{0},⊗) is not a group.
d. 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.
2.3 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.
2.4 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 .
2.7 Find all solutions in x of the equation system Ax = 12x,
where
A
and P3i=1 xi = 1.
2.8 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?
a. A = {(λ,λ + µ3,λ − µ3) | λ,µ ∈ R}
b. B = {(λ2,−λ2,0) | λ ∈ R}
c. Let γ be in R.
C = {(ξ1,ξ2,ξ3) ∈ R3 | ξ1 − 2ξ2 + 3ξ3 = γ}
d. 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
a. Determine the dimension of U1,U2.
b. Determine bases of U1 and U2.
c. Determine a basis of U1 ∩ U2.
2.14 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
2.15 Let F = {(x,y,z) ∈ R3 | x+y−z = 0} and G = {(a−b,a+b,a−3b) | a,b ∈ R}.
a. Show that F and G are subspaces of R3.
b. Calculate F ∩ G without resorting to any basis vector.
c. 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?
a. 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
x x
e. Let θ be in [0,2π[ and
Φ : R2 → R2
x x
2.17 Consider the linear mapping
Φ : R3 → R4
x1 3x1 + 2x2 + x3 x1 + x2 + x3
Φx2 = 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 .
a. Determine ker(Φ) and Im(Φ).
b. 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.
a. Show that B and B0 are two bases of R2 and draw those basis vectors.
b. Compute the matrix P1 that performs a basis change from B0 to B.
c. 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).
(i) Show that C is a basis of R3, e.g., by using determinants (see Section 4.1).
(ii) Let us call the standard basis of R3. Determine the matrix P2 that performs the basis change from C to C0.
d. 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.
e. Determine A0, the transformation matrix of Φ with respect to the bases
B0 and C0.
f. Let us consider the vector x ∈ R2 whose coordinates in B0 are [2,3]>.
In other words, x .
(i) Calculate the coordinates of x in B.
(ii) Based on that, compute the coordinates of Φ(x) expressed in C.
(iii) Then, write Φ(x) in terms of .
(iv) 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
a. hx,yi := x>y
b. hx,yi := x>Ay , A := 3.4 Compute the angle between
x , y
using
a. hx,yi := x>y
b. 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
a. Determine the orthogonal projection πU(x) of x onto U
b. 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.
