Linear Algebra
2.9 Further Reading
mappingΦ :V →W, anda∈W, the mapping
φ:V →W (2.132)
x7→a+ Φ(x) (2.133)
is anaffine mapping fromV toW. The vectorais called the translation affine mapping translation vector
vectorofφ.
Every affine mapping φ : V → W is also the composition of a linear mapping Φ : V → W and a translationτ : W → W inW, such that φ=τ ◦Φ. The mappingsΦandτ are uniquely determined.
The compositionφ0◦φof affine mappingsφ:V → W,φ0 :W →Xis affine.
Affine mappings keep the geometric structure invariant. They also pre-serve the dimension and parallelism.
2.9 Further Reading
There are many resources for learning linear algebra, including the text-books by Strang (2003), Golan (2007), Axler (2015), and Liesen and Mehrmann (2015). There are also several online resources that we men-tioned in the introduction to this chapter. We only covered Gaussian elim-ination here, but there are many other approaches for solving systems of linear equations, and we refer to numerical linear algebra textbooks by Stoer and Burlirsch (2002), Golub and Van Loan (2012), and Horn and Johnson (2013) for an in-depth discussion.
In this book, we distinguish between the topics of linear algebra (e.g., vectors, matrices, linear independence, basis) and topics related to the geometry of a vector space. In Chapter 3, we will introduce the inner product, which induces a norm. These concepts allow us to define angles, lengths and distances, which we will use for orthogonal projections. Pro-jections turn out to be key in many machine learning algorithms, such as linear regression and principal component analysis, both of which we will cover in Chapters 9 and 10, respectively.
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 Letnbe inN\{0}. Letk, xbe inZ. We define the congruence class¯kof the integerkas the set
k={x∈Z|x−k= 0 (modn)}
={x∈Z| ∃a∈Z: (x−k=n·a)}.
We now defineZ/nZ(sometimes writtenZn) as the set of all congruence classes modulon. Euclidean division implies that this set is a finite set con-tainingnelements:
Zn={0,1, . . . , n−1}
For alla, 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 allaandbinZnas
a⊗b=a×b , (2.135)
wherea×brepresents the usual multiplication inZ.
Letn= 5. Draw the times table of the elements ofZ5\{0}under⊗, i.e., calculate the productsa⊗bfor allaandbinZ5\{0}.
Hence, show that Z5\{0} is closed under ⊗ and possesses a neutral element for⊗. Display the inverse of all elements inZ5\{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 integersaandbare relatively prime (i.e.,gcd(a, b) = 1) if and only if there exist two integers uandvsuch thatau+bv= 1. Show that(Zn\{0},⊗)is a group if and only ifn∈N\{0}is prime.
2.3 Consider the setGof3×3matrices defined as follows:
G=
1 x z 0 1 y 0 0 1
∈R3×3
x, y, z∈R
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:
Exercises 65
2.5 Find the setS of all solutions inxof the following inhomogeneous linear systemsAx=b, whereAandbare defined as follows:
a.
2.6 Using Gaussian elimination, find all solutions of the inhomogeneous equa-tion systemAx=bwith
2.7 Find all solutions in x =
2.8 Determine the inverses of the following matrices if possible:
a. 2.9 Which of the following sets are subspaces ofR3?
a. A={(λ, λ+µ3, λ−µ3)|λ, µ∈R} b. B={(λ2,−λ2,0)|λ∈R}
c. Letγbe inR.
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. as linear combination of
x1=
Exercises 67 2.12 Consider two subspaces ofR4:
U1= span[
2.13 Consider two subspacesU1 and U2, whereU1 is the solution space of the homogeneous equation systemA1x=0andU2is the solution space of the homogeneous equation systemA2x=0with
A1=
2.14 Consider two subspacesU1andU2, whereU1is spanned by the columns of A1andU2is spanned by the columns ofA2with
b. CalculateF∩Gwithout resorting to any basis vector.
c. Find one basis forFand one forG, calculateF∩Gusing the basis vectors previously found and check your result with the previous question.
2.16 Are the following mappings linear?
a. Leta, b∈R.
whereL1([a, b])denotes the set of integrable functions on[a, b]. b.
Φ :C1→C0 f7→Φ(f) =f0,
where fork > 1,Ck denotes the set ofk times continuously differen-tiable functions, andC0denotes the set of continuous functions.
c.
2.17 Consider the linear mapping Φ :R3→R4 Find the transformation matrixAΦ.
Determinerk(AΦ).
Compute the kernel and image ofΦ. What aredim(ker(Φ))anddim(Im(Φ))? 2.18 LetEbe a vector space. Letfandgbe two automorphisms onEsuch that
f ◦g = idE (i.e.,f◦g is the identity mappingidE). Show thatker(f) = ker(g◦f),Im(g) = Im(g◦f)and thatker(f)∩Im(g) ={0E}.
2.19 Consider an endomorphism Φ : R3 → R3 whose transformation matrix (with respect to the standard basis inR3) is
AΦ= a. Determineker(Φ)andIm(Φ).
b. Determine the transformation matrixA˜Φwith respect to the basis B= ( i.e., perform a basis change toward the new basisB.
2.20 Let us considerb1,b2,b01,b02,4vectors ofR2expressed in the standard basis
Exercises 69 a. Show thatBandB0 are two bases ofR2and draw those basis vectors.
b. Compute the matrixP1that performs a basis change fromB0toB. c. We considerc1,c2,c3, three vectors ofR3defined in the standard basis
ofR3as
c1=
1 2
−1
, c2=
0
−1 2
, c3=
1 0
−1
and we defineC= (c1,c2,c3).
(i) Show that C is a basis of R3, e.g., by using determinants (see Section 4.1).
(ii) Let us callC0 = (c01,c02,c03)the standard basis ofR3. Determine the matrixP2that performs the basis change fromCtoC0. d. We consider a homomorphismΦ :R2−→R3, such that
Φ(b1+b2) = c2+c3
Φ(b1−b2) = 2c1−c2+ 3c3
whereB= (b1,b2)andC= (c1,c2,c3)are ordered bases ofR2andR3, respectively.
Determine the transformation matrixAΦ of Φwith respect to the or-dered basesB andC.
e. DetermineA0, the transformation matrix ofΦwith respect to the bases B0andC0.
f. Let us consider the vectorx∈R2 whose coordinates inB0 are[2,3]>. In other words,x= 2b01+ 3b02.
(i) Calculate the coordinates ofxinB.
(ii) Based on that, compute the coordinates ofΦ(x)expressed inC. (iii) Then, writeΦ(x)in terms ofc01,c02,c03.
(iv) Use the representation ofxinB0 and the matrixA0 to find this result directly.