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Linear algebra

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces, often represented by matrices. Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations. Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer science, computer animation, and the social sciences (particularly in economics). Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear ones. .^[1]^[2]

. Wilson Kaye, Richard Kaye and Rob Wilson

Covers determinants, linear spaces, systems of linear equations, linear functions of a vector argument, coordinate transformations, the canonical form of the matrix of a linear operator, bilinear and quadratic forms, Euclidean spaces, unitary spaces, quadratic forms in Euclidean and unitary spaces, finite-dimensional space. Problems with hints and answers. Georgi E. Shilov (Author), Richard A. Silverman (Editor)

This text for a second course in linear algebra, aimed at math majors and graduates, adopts a novel approach by banishing determinants to the end of the book and focusing on understanding the structure of linear operators on vector spaces. The author has taken unusual care to motivate concepts and to simplify proofs. For example, the book presents - without having defined determinants - a clean proof that every linear operator on a finite-dimensional complex vector space has an eigenvalue. The book starts by discussing vector spaces, linear independence, span, basics, and dimension. Students are introduced to inner-product spaces in the first half of the book and shortly thereafter to the finite- dimensional spectral theorem. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. This second edition features new chapters on diagonal matrices, on linear functionals and adjoints, and on the spectral theorem; some sections, such as those on self-adjoint and normal operators, have been entirely rewritten; and hundreds of minor improvements have been made throughout the text. Sheldon Axler

Affine space

In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space it is possible to talk of vectors representing displacements ("subtractions") between points, and to use vectors to describe translations and parallelism. The vectors of affine space generalise the notion of displacement vector. As for Euclidean space, any point in an affine space can be chosen as an origin. Any other point can then be uniquely identified by a vector from the origin, generalising the notion of a position vector. An affine space allows us describe transformations of vector space (e.g stretch, rotation, shear) centred at any point, and adds translations to those maps.

Informal description

In the words of the French mathematician Marcel Berger, "An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps"^[3]. Starting with a vector space $V$ we introduce a set $A$ of points in one-one correspondence with $V$ . The origin is the point in $A$ corresponding to the zero vector of $V$ . We now introduce translations on $A$ such that any other point could equally well serve as the origin. By convention, vectors are shown in bold font.

Imagine that Alice uses point $a$ as the origin, but Bob is using another point, $b$ , as the origin. Two vectors, $p$ and $q$ , are to be added. Bob draws an arrow from $b$ to point $b + p$ and another arrow from point $b + p$ to $b + p + q$ . Bob completes the parallelogram to find the diagonal, which is $p + q$ because, for Bob, $b$ corresponds to the zero vector. Meanwhile, Alice draws an arrow from $a$ to point $a + p$ and another arrow from point $a + p$ to $a + p + q$ . Alice completes her parallelogram, and also finds $p + q$ because, for Alice, $a$ corresponds to the zero vector. If $b$ is such that $b$ = $a + b$ then Alice might say that Bob has worked out $a + b + p + q - b$ = $p + q$ . Regardless of their chosen origin, both Alice and Bob obtain the same result for the vector addition, since parallel vectors of equal magnitude are equal.

Definition

An affine space^[4] is a set $A$ together with a vector space $V$ over a field $F$ and a faithful and transitive group action of $V$ (with addition of vectors as group action) on $A$ .

Explicitly, an affine space is a point set $A$ together with a map

l\colon V\times A\to A,\;(\mathbf {v} ,a)\mapsto \mathbf {v} +a

with the following properties:.^[5]Snapper, Ernst; Troyer, Robert J. (1989), Metric Affine Geometry, p. 6</ref>^[6]

Left identity
$\forall a\in A,\;\mathbf {0} +a=a$
Associativity
$\forall \mathbf {v} ,\mathbf {w} \in V,\forall a\in A,\;\mathbf {v} +(\mathbf {w} +a)=(\mathbf {v} +\mathbf {w} )+a$
Uniqueness
$\forall a\in A,\;V\to A\colon \mathbf {v} \mapsto \mathbf {v} +a\quad$ is a bijection.

It is also possible to place vectors on the right.^[7] $F$ is called the coefficient field.

The vector space $V$ is said to underlie the affine space $A$ and is also called the difference space. Any vector space, $V$ , can be regarded as an affine space over itself.

The uniqueness property ensures that subtraction of any two elements of $A$ is well defined, producing a vector of $V$ . By noting that one can define subtraction of points of an affine space as follows:

{\overrightarrow {ab}}\;\equiv \;a\,-\,b\;

is the unique vector in

V

such that

\left(a\,-\,b\right)\,+\,b\;=\;a

,

one can equivalently define an affine space as a point set $A$ , together with a vector space $V$ , and a subtraction map

\operatorname {\phi } :\;A\,\times \,A\;\to \;V,\;\left(a,\,b\right)\,\mapsto \,b\,-\,a\;\equiv \;{\overrightarrow {ab}}

with the following properties:^[8]

$\forall p\,\in \,A,\;\forall \mathbf {v} \,\in \,V$ there is a unique point $q\,\in \,A$ such that $q\,-\,p\;=\;\mathbf {v}$ and
$\forall p,\,q,\,r\,\in \,A,\;(q\,-\,p)\,+\,(r\,-\,q)\;=\;r\,-\,p$ .

These two properties are called Weyl's axioms.

Examples

When children find the answers to sums such as 4 + 3 or 4 − 2 by counting right or left on a number line, they are treating the number line as a one-dimensional affine space.
Any coset of a subspace $V$ of a vector space is an affine space over that subspace.
If $T$ is a matrix and $b$ lies in its column space, the set of solutions of the equation $T x = b$ is an affine space over the subspace of solutions of $T x = 0$ .
The solutions of an inhomogeneous linear differential equation form an affine space over the solutions of the corresponding homogeneous linear equation.
Generalizing all of the above, if $T : V \to W$ is a linear mapping and $y$ lies in its image, the set of solutions $x \in V$ to the equation $T x = y$ is a coset of the kernel of $T$ , and is therefore an affine space over $Ker T$ .

Affine subspaces

An affine subspace (sometimes called a linear manifold, linear variety, or a flat) of a vector space $V$ is a subset closed under affine combinations of vectors in the space. For example, the set

A={\Bigl \{}\sum _{i}^{N}\alpha _{i}\mathbf {v} _{i}{\Big |}\sum _{i}^{N}\alpha _{i}=1{\Bigr \}}

is an affine space, where $\scriptstyle \{\mathbf {v} _{i}\}_{i\in I}$ is a family of vectors in $V$ ; this space is the affine span of these points. To see that this is indeed an affine space, observe that this set carries a transitive action of the vector subspace $W$ of $V$

W={\Bigl \{}\sum _{i}^{N}\beta _{i}\mathbf {v} _{i}{\Big |}\sum _{i}^{N}\beta _{i}=0{\Bigr \}}.

This affine subspace can be equivalently described as the coset of the $W$ -action

S=\mathbf {p} +W,\,

where $p$ is any element of $A$ , or equivalently as any level set of the quotient map $V \to V / W$ . A choice of $p$ gives a base point of $A$ and an identification of $W$ with $A$ , but there is no natural choice, nor a natural identification of $W$ with $A$ .

A linear transformation is a function that preserves all linear combinations; an affine transformation is a function that preserves all affine combinations. A linear subspace is an affine subspace containing the origin, or, equivalently, a subspace that is closed under linear combinations.

For example, in $\scriptstyle {\mathbb {R} ^{3}}$ , the origin, lines and planes through the origin and the whole space are linear subspaces, while points, lines and planes in general as well as the whole space are the affine subspaces.

Affine combinations and affine dependence

An affine combination is a linear combination in which the sum of the coefficients is 1. Just as members of a set of vectors are linearly independent if none is a linear combination of the others, so also they are affinely independent if none is an affine combination of the others. The set of linear combinations of a set of vectors is their "linear span" and is always a linear subspace; the set of all affine combinations is their "affine span" and is always an affine subspace. For example, the affine span of a set of two points is the line that contains both; the affine span of a set of three non-collinear points is the plane that contains all three.

Vectors

v 1, v 2, \dots , v n

are linearly dependent if there exist scalars $a 1, a 2, \dots , a n$ , not all zero, for which

a 1 v 1 + a 2 v 2 + \dots + a n v n = 0

(1)

Similarly they are affinely dependent if in addition the sum of coefficients is zero:

\sum _{i=1}^{n}a_{i}=0.

Geometric objects as points and vectors

In an affine space, geometric objects have two different (although related) descriptions on languages of points (elements of $A$ ) and vectors (elements of $V$ ). A vector description can specify an object only up to translations.

Geometry	Points	Vectors
A point	One point P	none (zero vector space)
A line (1-subspace)	Can be specified with two distinct points	A non-zero vector up to multiplication to (non-zero) scalars
A line segment	Two (independent) points: P, Q	One vector P Q→, or two dependent (mutually opposite) vectors P Q→ and Q P→
A plane (2-subspace)	Can be specified with three points not lying on one line	A linear 2-subspace, can be specified with two linearly-independent vectors
A triangle	Three (independent) points: △P Q R	Three dependent vectors related as P R→ = P Q→ + Q R→, or P Q→ + Q R→ + R P→ = 0, or just two independent vectors
A parallelogram	Four points: ▱P Q R S of which any three determine the fourth	Two independent vectors: P Q→ = S R→ P S→ = Q R→

Axioms

Affine space is usually studied as analytic geometry using coordinates, or equivalently vector spaces. It can also be studied as synthetic geometry by writing down axioms, though this approach is much less common. There are several different systems of axioms for affine space.

Coxeter (1969, p. 192) axiomatizes affine geometry (over the reals) as ordered geometry together with an affine form of Desargues's theorem and an axiom stating that in a plane there is at most one line through a given point not meeting a given line.

Affine planes satisfy the following axioms (Cameron 1991, chapter 2): (in which two lines are called parallel if they are equal or disjoint):

Any two distinct points lie on a unique line.
Given a point and line there is a unique line which contains the point and is parallel to the line
There exist three non-collinear points.

As well as affine planes over fields (or division rings), there are also many non-Desarguesian planes satisfying these axioms. (Cameron 1991, chapter 3) gives axioms for higher-dimensional affine spaces.

Relation to projective spaces

Affine spaces are subspaces of projective spaces: an affine plane can be obtained from any projective plane by removing a line and all the points on it, and conversely any affine plane can be used to construct a projective plane as a closure by adding a line at infinity whose points correspond to equivalence classes of parallel lines.

Further, transformations of projective space that preserve affine space (equivalently, that leave the hyperplane at infinity invariant as a set) yield transformations of affine space. Conversely, any affine linear transformation extends uniquely to a projective linear transformations, so the affine group is a subgroup of the projective group. For instance, Möbius transformations (transformations of the complex projective line, or Riemann sphere) are affine (transformations of the complex plane) if and only if they fix the point at infinity.

However, one cannot take the projectivization of an affine space, so projective spaces are not naturally quotients of affine spaces: one can only take the projectivization of a vector space, since the projective space is lines through a given point, and there is no distinguished point in an affine space. If one chooses a base point (as zero), then an affine space becomes a vector space, which one may then projectivize, but this requires a choice.

Notes

^ Strang, Gilbert (July 19, 2005), Linear Algebra and Its Applications (4th ed.), Brooks Cole, ISBN 978-0-03-010567-8
^ Weisstein, Eric. "Linear Algebra". From MathWorld--A Wolfram Web Resource. Wolfram. Retrieved 16 April 2012.
^ Berger 1987, p. 32
^ Berger, Marcel (1984), "Affine spaces", Problems in Geometry, p. 11, ISBN 9780387909714
^ Berger 1987, p. 33
^ Tarrida, Agusti R. (2011), "Affine spaces", Affine Maps, Euclidean Motions and Quadrics, pp. 1–2, ISBN 9780857297105
^ Weisstein, Eric. "Affine Space". From MathWorld--A Wolfram Web Resource. Wolfram. Retrieved 16 July 2014.
^ Nomizu & Sasaki 1994, p. 7

References

Berger, Marcel (1984), "Affine spaces", Problems in Geometry, Springer-Verlag, ISBN 978-0-387-90971-4
Berger, Marcel (1987), Geometry I, Berlin: Springer, ISBN 3-540-11658-3
Cameron, Peter J. (1991), Projective and polar spaces, QMW Maths Notes, vol. 13, London: Queen Mary and Westfield College School of Mathematical Sciences, MR 1153019
Coxeter, Harold Scott MacDonald (1969), Introduction to Geometry (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-50458-0, MR 0123930
Dolgachev, I.V.; Shirokov, A.P. (2001) [1994], "RQG/sandbox", Encyclopedia of Mathematics, EMS Press
Snapper, Ernst; Troyer, Robert J. (1989), Metric Affine Geometry (Dover edition, first published in 1989 ed.), Dover Publications, ISBN 0-486-66108-3
Nomizu, K.; Sasaki, S. (1994), Affine Differential Geometry (New ed.), Cambridge University Press, ISBN 978-0-521-44177-3
Tarrida, Agusti R. (2011), "Affine spaces", Affine Maps, Euclidean Motions and Quadrics, Springer, ISBN 978-0-85729-709-9

Category:Affine geometry Category:Linear algebra

[1] Strang, Gilbert (July 19, 2005), Linear Algebra and Its Applications (4th ed.), Brooks Cole, ISBN 978-0-03-010567-8

[2] Weisstein, Eric. "Linear Algebra". From MathWorld--A Wolfram Web Resource. Wolfram. Retrieved 16 April 2012.

[3] Berger 1987, p. 32

[4] Berger, Marcel (1984), "Affine spaces", Problems in Geometry, p. 11, ISBN 9780387909714

[Berger1987-5] Berger 1987, p. 33

[6] Tarrida, Agusti R. (2011), "Affine spaces", Affine Maps, Euclidean Motions and Quadrics, pp. 1–2, ISBN 9780857297105

[7] Weisstein, Eric. "Affine Space". From MathWorld--A Wolfram Web Resource. Wolfram. Retrieved 16 July 2014.

[8] Nomizu & Sasaki 1994, p. 7

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