Linear Algebra I [Lecture notes]

225 Pages·2016·1.293 MB·English

by  Ronald van Luijk

Linear Algebra I
Ronald van Luijk, 2016
With many parts from “Linear Algebra I” by Michael Stoll, 2007

Contents
Dependencies among sections 3
Chapter 1. Euclidean space: lines and hyperplanes 5
1.1. Definition 5
1.2. Euclidean plane and Euclidean space 6
1.3. The standard scalar product 9
1.4. Angles, orthogonality, and normal vectors 13
1.5. Orthogonal projections and normality 19
1.5.1. Projecting onto lines and hyperplanes containing zero 19
1.5.2. Projecting onto arbitrary lines and hyperplanes 23
1.6. Distances 25
1.7. Reflections 29
1.7.1. Reflecting in lines and hyperplanes containing zero 30
1.7.2. Reflecting in arbitrary lines and hyperplanes 32
1.8. Cauchy-Schwarz 34
1.9. What is next? 36
Chapter 2. Vector spaces 39
2.1. Definition of a vector space 40
2.2. Examples 41
2.3. Basic properties 47
Chapter 3. Subspaces 51
3.1. Definition and examples 51
3.2. The standard scalar product (again) 54
3.3. Intersections 56
3.4. Linear hulls, linear combinations, and generators 57
3.5. Sums of subspaces 63
Chapter 4. Linear maps 69
4.1. Definition and examples 69
4.2. Linear maps form a vector space 74
4.3. Characterising linear maps 79
4.4. Isomorphisms 81
Chapter 5. Matrices 85
5.1. Definition of matrices 85
5.2. Matrix associated to a linear map 86
5.3. The product of a matrix and a vector 88
5.4. Linear maps associated to matrices 89
5.5. Addition and multiplication of matrices 91
5.6. Row space, column space, and transpose of a matrix 97
Chapter 6. Computations with matrices 101
6.1. Elementary row and column operations 101
1
2 CONTENTS
6.2. Row echelon form 104
6.3. Generators for the kernel 109
6.4. Reduced row echelon form 112
Chapter 7. Linear independence and dimension 115
7.1. Linear independence 115
7.2. Bases 121
7.3. The basis extension theorem and dimension 127
7.4. Dimensions of subspaces 135
Chapter 8. Ranks 139
8.1. The rank of a linear map 139
8.2. The rank of a matrix 142
8.3. Computing intersections 146
8.4. Inverses of matrices 148
Chapter 9. Linear maps and matrices 153
9.1. The matrix associated to a linear map 153
9.2. The matrix associated to the composition of linear maps 157
9.3. Changing bases 159
9.4. Endomorphisms 161
9.5. Similar matrices and the trace 162
9.6. Classifying matrices 163
9.6.1. Similar matrices 163
9.6.2. Equivalent matrices 165
Chapter 10. Determinants 169
10.1. Determinants of matrices 169
10.2. Some properties of the determinant 176
10.3. Cramer’s rule 179
10.4. Determinants of endomorphisms 179
10.5. Linear equations 181
Chapter 11. Eigenvalues and Eigenvectors 187
11.1. Eigenvalues and eigenvectors 187
11.2. The characteristic polynomial 189
11.3. Diagonalization 193
Appendix A. Review of maps 203
Appendix B. Fields 205
B.1. Definition of fields 205
B.2. The field of complex numbers. 207
Appendix C. Labeled collections 209
Appendix D. Infinite-dimensional vector spaces and Zorn’s Lemma 211
Bibliography 215
Index of notation 217
Index 219
Dependencies among sections
11.1-11.3
10.3 10.5
10.1,10.2,10.4
9.6
9.1-9.5
8.3 8.4
8.1,8.2
7.4
7.1-7.3 6.4
6.1-6.3
5.1-5.6 4.4
3.5 4.1-4.3
3.1-3.4
2.1-2.3
1.7.2 1.9
1.5.2 1.7.1 1.6 1.8
1.5.1
1.1-1.4
3

CHAPTER 1
Euclidean space: lines and hyperplanes
This chapter deals, for any non-negative integer n, with Euclidean n-space Rn,
which is the set of all (ordered) sequences of n real numbers, together with a
distance that we will define. We make it slightly more general, so that we can
also apply our theory to, for example, the rational numbers instead of the real
numbers: instead of just the set R of real numbers, we consider any subfield of R.
At this stage, it suffices to say that a subfield of R is a nonempty subset F ⊂ R
containing 0 and 1, in which we can multiply, add, subtract, and divide (except
by 0); that is, for any x,y ∈ F, also the elements xy,x + y,x − y (and x/y if
y (cid:54)= 0) are contained in F. We refer the interested reader to Appendix B for a
more precise definition of a field in general.
Therefore, for this entire chapter (and only this chapter), we let F denote a sub-
fieldofR, suchasthefieldRitselforthefieldQofrationalnumbers. Furthermore,
we let n denote a non-negative integer.
1.1. Definition
An n-tuple is an ordered sequence of n objects. We let Fn denote the set of all
n-tuples of elements of F. For example, the sequence
√
(cid:0) (cid:1)
−17,0,3,1+ 2,eπ
is an element of R5. The five numbers are separated by commas. In general, if we
have n numbers x ,x ,...,x ∈ F, then
1 2 n
x = (x ,x ,...,x )
1 2 n
is an element of Fn. Again, the numbers are separated by commas. Such n-tuples
are called vectors; the numbers in a vector are called coordinates. In other words,
the i-th coordinate of the vector x = (x ,x ,...,x ) is the number x .
1 2 n i
We define an addition by adding two elements of Fn coordinate-wise:
(x ,x ,...,x )⊕(y ,y ,...,y ) = (x +y ,x +y ,...,x +y ).
1 2 n 1 2 n 1 1 2 2 n n
For example, the sequence (12,14,16,18,20,22,24) is an element of R7 and we
have
(12,14,16,18,20,22,24)+(11,12,13,14,13,12,11) = (23,26,29,32,33,34,35).
Unsurprisingly, we also define a coordinate-wise subtraction:
(x ,x ,...,x )(cid:9)(y ,y ,...,y ) = (x −y ,x −y ,...,x −y ).
1 2 n 1 2 n 1 1 2 2 n n
Until the end of this section, we denote the sum and the difference of two elements
x,y ∈ Fn by x⊕y and x(cid:9)y, respectively, in order to distinguish them from the
usual addition and subtraction of two numbers. Similarly, we define a scalar
multiplication: for any element λ ∈ F, we set
λ(cid:12)(x ,x ,...,x ) = (λx ,λx ,...,λx ).
1 2 n 1 2 n
5
6 1. EUCLIDEAN SPACE: LINES AND HYPERPLANES
This is called scalar multiplication because the elements of Fn are scaled; the
elements of F, by which we scale, are often called scalars. We abbreviate the
special vector (0,0,...,0) consisting of only zeros by 0, and for any vector x ∈ Fn,
we abbreviate the vector 0(cid:9)x by −x. In other words, we have
−(x ,x ,...,x ) = (−x ,−x ,...,−x ).
1 2 n 1 2 n
Because our new operations are all defined coordinate-wise, they obviously satisfy
the following properties:
(1) for all x,y ∈ Fn, we have x⊕y = y ⊕x;
(2) for all x,y,z ∈ Fn, we have (x⊕y)⊕z = x⊕(y ⊕z);
(3) for all x ∈ Fn, we have 0⊕x = x and 1(cid:12)x = x;
(4) for all x ∈ Fn, we have (−1)(cid:12)x = −x and x⊕(−x) = 0;
(5) for all x,y,z ∈ Fn, we have x⊕y = z if and only if y = z (cid:9)x;
(6) for all x,y ∈ Fn, we have x(cid:9)y = x⊕(−y);
(7) for all λ,µ ∈ F and x ∈ Fn, we have λ(cid:12)(µ(cid:12)x) = (λ·µ)(cid:12)x;
(8) for all λ,µ ∈ F and x ∈ Fn, we have (λ+µ)(cid:12)x = (λ(cid:12)x)⊕(µ(cid:12)x);
(9) for all λ ∈ F and x,y ∈ Fn, we have λ(cid:12)(x⊕y) = (λ(cid:12)x)⊕(λ(cid:12)y).
In fact, in the last two properties, we may also replace + and ⊕ by − and (cid:9),
respectively, but the properties that we then obtain follow from the properties
above. All these properties together mean that the operations ⊕, (cid:9), and (cid:12) really
behave like the usual addition, subtraction, and multiplication, as long as we
remember that the scalar multiplication is a multiplication of a scalar
with a vector, and not of two vectors!.
We therefore will usually leave out the circle in the notation: instead of x⊕y and
x(cid:9)y we write x+y and x−y, and instead of λ(cid:12)x we write λ·x or even λx.
As usual, scalar multiplication takes priority over addition and subtraction, so
when we write λx±µy with λ,µ ∈ F and x,y ∈ Fn, we mean (λx)±(µy). Also
as usual, when we have t vectors x ,x ,...,x ∈ Fn, the expression x ±x ±x ±
1 2 t 1 2 3
···±x should be read from left to right, so it stands for
t
(...((x ±x )±x )±···)±x .
1 2 3 t
(cid:124) (cid:123)(cid:122) (cid:125)
t−2
If all the signs in the expression are positive (+), then any other way of putting
the parentheses would yield the same by property (2) above.
1.2. Euclidean plane and Euclidean space
For n = 2 or n = 3 we can identify Rn with the pointed plane or three-dimensional
space, respectively. We say pointed because they come with a special point,
namely 0. For instance, for R2 we take an orthogonal coordinate system in the
plane, with 0 at the origin, and with equal unit lengths along the two coordinate
axes. Then the vector p = (p ,p ) ∈ R2, which is by definition nothing but a pair
1 2
of real numbers, corresponds with the point in the plane whose coordinates are p
1
and p . In this way, the vectors get a geometric interpretation. We can similarly
2
identify R3 with three-dimensional space. We will often make these identifications
and talk about points as if they are vectors, and vice versa. By doing so, we can
now add points in the plane, as well as in space! Figure 1.1 shows the two points
p = (3,1) and q = (1,2) in R2, as well as the points 0,−p,2p,p+q, and q −p.
For n = 2 or n = 3, we may also represent vectors by arrows in the plane or
space, respectively. In the plane, the arrow from the point p = (p ,p ) to the
1 2
1.2. EUCLIDEAN PLANE AND EUCLIDEAN SPACE 7
p+q
q
2p
q −p
p
0
−p
Figure 1.1. Two points p and q in R2, as well as 0,−p,2p,p+q,
and q −p
point q = (q ,q ) represents the vector v = (q −p ,q −p ) = q −p. (A careful
1 2 1 1 2 2
reader notes that here we do indeed identify points and vectors.) We say that the
point p is the tail of the arrow and the point q is the head. Note the distinction
we make between an arrow and a vector, the latter of which is by definition just a
sequence of real numbers. Many different arrows may represent the same vector v,
but all these arrows have the same direction and the same length, which together
narrow down the vector. One arrow is special, namely the one with 0 as its tail;
the head of this arrow is precisely the point q − p, which is the point identified
with v! See Figure 1.2, in which the arrows are labeled by the name of the vector v
they represent, and the points are labeled either by their own name (p and q), or
the name of the vector they correspond with (v or 0). Note that besides v = q−p,
we (obviously) also have q = p+v.
q
v
q −p = v
v p
0
Figure 1.2. Two arrows representing the same vector v = (−2,1)
Of course we can do the same for R3. For example, take the two points p =
(3,1,−4) and q = (−1,2,1) and set v = q −p. Then we have v = (−4,1,5). The
arrow from p to q has the same direction and length as the arrow from 0 to the
point (−4,1,5). Both these arrows represent the vector v.
Note that we now have three notions: points, vectors, and arrows.
points vectors arrows
Vectors and points can be identified with each other, and arrows represent vectors
(and thus points).
We can now interpret negation, scalar multiples, sums, and differences of vectors
(as defined in Section 1.1) geometrically, namely in terms of points and arrows.
8 1. EUCLIDEAN SPACE: LINES AND HYPERPLANES
For points this was already depicted in Figure 1.1. If p is a point in R2, then
−p is obtained from p by rotating it 180 degrees around 0; for any real number
λ > 0, the point λp is on the half line from 0 through p with distance to 0 equal
to (λ times the distance from p to 0). For any points p and q in R2 such that 0,p,
and q are not collinear, the points p+q and q−p are such that the four points 0,
p, p+q, and q are the vertices of a parallelogram with p and q opposite vertices,
and the four points 0, −p, q−p, q are the vertices of a parallelogram with −p and
q opposite vertices.
In terms of arrows we get the following. If a vector v is represented by a certain
arrow, then −v is represented by any arrow with the same length but opposite
direction; furthermore, for any positive λ ∈ R, the vector λv is represented by the
arrow obtained by scaling the arrow representing v by a factor λ.
If v and w are represented by two arrows that have common tail p, then these two
arrows are the sides of a unique parallelogram; the vector v + w is represented
by a diagonal in this parallelogram, namely the arrow that also has p as tail and
whose head is the opposite point in the parallelogram. An equivalent description
for v+w is to take two arrows, for which the head of the one representing v equals
the tail of the one representing w; then v + w is represented by the arrow from
the tail of the first to the head of the second. See Figure 1.3.
w v
v +w
q
v −w
r
−w
v
w
v +(−w)
p
v −w
Figure 1.3. Geometric interpretation of addition and subtraction
The description of laying the arrows head-to-tail generalises well to the addition of
more than two vectors. Let v ,v ,...,v in R2 or R3 be vectors and p ,p ,...,p
1 2 t 0 1 t
points such that v is represented by the arrow from p to p . Then the sum
i i−1 i
v +v +···+v is represented by the arrow from p to p . See Figure 1.4.
1 2 t 0 t
For the same v and w, still represented by arrows with common tail p and with
headsq andr, respectively, thedifferencev−w isrepresentedbytheotherdiagonal
in the same parallelogram, namely the arrow from r to q. Another construction
for v−w is to write this difference as the sum v+(−w), which can be constructed
as described above. See Figure 1.3.
Representing vectors by arrows is very convenient in physics. In classical mechan-
ics, for example, we identify forces applied on a body with vectors, which are often
depicted by arrows. The total force applied on a body is then the sum of all the
forces in the sense that we have defined it.
Motivated by the case n = 2 and n = 3, we will sometimes refer to vectors in
Rn as points in general. Just as arrows in R2 and R3 are uniquely determined by
their head and tail, for general n we define an arrow to be a pair (p,q) of points
p,q ∈ Rn and we say that this arrow represents the vector q−p. The points p and
q are the tail and the head of the arrow (p,q).