MATRICES

 

Matrices[edit]

Main articles: Matrix and Determinant

A typical matrix

Matrices are a useful notion to encode linear maps.^[22] They are written as a rectangular array of scalars as in the image at the right. Any m-by-n matrix $�$ gives rise to a linear map from Fⁿ to F^m, by the following

$${\displaystyle \mathbf{�}=({�}_1,{�}_2,\dots ,{�}_{�})\mapsto \left(\sum _{�=1}^{�}{�}_{1�}{�}_{�},\sum _{�=1}^{�}{�}_{2�}{�}_{�},\dots ,\sum _{�=1}^{�}{�}_{��}{�}_{�}\right),}$$

where $\sum$ denotes summation, or, using the matrix multiplication of the matrix $�$ with the coordinate vector ${\displaystyle \mathbf{�}:}$

$${\displaystyle \mathbf{�}\mapsto �\mathbf{�}.}$$

Moreover, after choosing bases of V and W, any linear map f : V → W is uniquely represented by a matrix via this assignment.^[23]

The volume of this parallelepiped is the absolute value of the determinant of the 3-by-3 matrix formed by the vectors r₁, r₂, and r₃.

The determinant det (A) of a square matrix A is a scalar that tells whether the associated map is an isomorphism or not: to be so it is sufficient and necessary that the determinant is nonzero.^[24] The linear transformation of Rⁿ corresponding to a real n-by-n matrix is orientation preserving if and only if its determinant is positive.

Eigenvalues and eigenvectors[edit]

Main article: Eigenvalues and eigenvectors

Endomorphisms, linear maps f : V → V, are particularly important since in this case vectors v can be compared with their image under f, f(v). Any nonzero vector v satisfying λv = f(v), where λ is a scalar, is called an eigenvector of f with eigenvalue λ.^{[nb 4][25]} Equivalently, v is an element of the kernel of the difference f − λ · Id (where Id is the identity map V → V). If V is finite-dimensional, this can be rephrased using determinants: f having eigenvalue λ is equivalent to

$${\displaystyle det(�-�\cdot \mathrm{Id})=0.}$$

By spelling out the definition of the determinant, the expression on the left hand side can be seen to be a polynomial function in λ, called the characteristic polynomial of f.^[26] If the field F is large enough to contain a zero of this polynomial (which automatically happens for F algebraically closed, such as F = C) any linear map has at least one eigenvector. The vector space V may or may not possess an eigenbasis, a basis consisting of eigenvectors. This phenomenon is governed by the Jordan canonical form of the map.^{[27][nb 5]} The set of all eigenvectors corresponding to a particular eigenvalue of f forms a vector space known as the eigenspace corresponding to the eigenvalue (and f) in question. To achieve the spectral theorem, the corresponding statement in the infinite-dimensional case, the machinery of functional analysis is needed, see below.^{[clarification needed]}

Basic constructions[edit]

In addition to the above concrete examples, there are a number of standard linear algebraic constructions that yield vector spaces related to given ones. In addition to the definitions given below, they are also characterized by universal properties, which determine an object $�$ by specifying the linear maps from $�$ to any other vector space.

Subspaces and quotient spaces[edit]

Main articles: Linear subspace and Quotient vector space

A line passing through the origin (blue, thick) in R³ is a linear subspace. It is the intersection of two planes (green and yellow).

A nonempty subset $�$ of a vector space $�$ that is closed under addition and scalar multiplication (and therefore contains the $\mathbf{0}$-vector of $�$) is called a linear subspace of $�,$ or simply a subspace of $�,$ when the ambient space is unambiguously a vector space.^{[28][nb 6]} Subspaces of $�$ are vector spaces (over the same field) in their own right. The intersection of all subspaces containing a given set $�$ of vectors is called its span, and it is the smallest subspace of $�$ containing the set $�.$Expressed in terms of elements, the span is the subspace consisting of all the linear combinations of elements of $�.$^[29]

A linear subspace of dimension 1 is a vector line. A linear subspace of dimension 2 is a vector plane. A linear subspace that contains all elements but one of a basis of the ambient space is a vector hyperplane. In a vector space of finite dimension ${\displaystyle �,}$ a vector hyperplane is thus a subspace of dimension ${\displaystyle �-1.}$

The counterpart to subspaces are quotient vector spaces.^[30] Given any subspace ${\displaystyle �\subseteq �,}$ the quotient space ${\displaystyle �/�}$ ("$�$ modulo $�$") is defined as follows: as a set, it consists of ${\displaystyle \mathbf{�}+�=\{\mathbf{�}+\mathbf{�}:\mathbf{�}\in �\},}$ where $\mathbf{�}$ is an arbitrary vector in $�.$ The sum of two such elements ${\displaystyle {\mathbf{�}}_1+�}$ and ${\displaystyle {\mathbf{�}}_2+�}$ is ${\displaystyle \left({\mathbf{�}}_1+{\mathbf{�}}_2\right)+�,}$ and scalar multiplication is given by ${\displaystyle �\cdot (\mathbf{�}+�)=(�\cdot \mathbf{�})+�.}$ The key point in this definition is that ${\displaystyle {\mathbf{�}}_1+�={\mathbf{�}}_2+�}$ if and only if the difference of ${\mathbf{�}}_1$ and ${\mathbf{�}}_2$ lies in $�.$^{[nb 7]} This way, the quotient space "forgets" information that is contained in the subspace $�.$

The kernel ${\displaystyle \ker (�)}$ of a linear map $�:�\to �$ consists of vectors $\mathbf{�}$ that are mapped to $\mathbf{0}$ in $�.$^[31] The kernel and the image ${\displaystyle \mathrm{im}(�)=\{�(\mathbf{�}):\mathbf{�}\in �\}}$ are subspaces of $�$ and $�,$ respectively.^[32] The existence of kernels and images is part of the statement that the category of vector spaces (over a fixed field $�$) is an abelian category, that is, a corpus of mathematical objects and structure-preserving maps between them (a category) that behaves much like the category of abelian groups.^[33] Because of this, many statements such as the first isomorphism theorem (also called rank–nullity theorem in matrix-related terms)

$${\displaystyle �/\ker (�)\equiv \mathrm{im}(�)}$$

and the second and third isomorphism theorem can be formulated and proven in a way very similar to the corresponding statements for groups.

An important example is the kernel of a linear map ${\displaystyle \mathbf{�}\mapsto �\mathbf{�}}$ for some fixed matrix $�,$ as above.^{[clarification needed]} The kernel of this map is the subspace of vectors $\mathbf{�}$ such that ${\displaystyle �\mathbf{�}=\mathbf{0},}$ which is precisely the set of solutions to the system of homogeneous linear equations belonging to $�.$ This concept also extends to linear differential equations

$${\displaystyle {�}_0�+{�}_1\frac{��}{��}+{�}_2\frac{{�}^2�}{�{�}^2}+\cdots +{�}_{�}\frac{{�}^{�}�}{�{�}^{�}}=0,}$$

where the coefficients ${�}_{�}$ are functions in $�,$ too. In the corresponding map

$${\displaystyle �\mapsto �(�)=\sum _{�=0}^{�}{�}_{�}\frac{{�}^{�}�}{�{�}^{�}},}$$

the derivatives of the function $�$ appear linearly (as opposed to ${\displaystyle {�}^{\mathrm{\prime }\mathrm{\prime }}(�{)}^2,}$ for example). Since differentiation is a linear procedure (that is, ${\displaystyle (�+�{)}^{\mathrm{\prime }}={�}^{\mathrm{\prime }}+{�}^{\mathrm{\prime }}}$ and ${\displaystyle (�\cdot �{)}^{\mathrm{\prime }}=�\cdot {�}^{\mathrm{\prime }}}$ for a constant $�$) this assignment is linear, called a linear differential operator. In particular, the solutions to the differential equation ${\displaystyle �(�)=0}$ form a vector space (over R or C).

Direct product and direct sum[edit]

Main articles: Direct product and Direct sum of modules

The direct product of vector spaces and the direct sum of vector spaces are two ways of combining an indexed family of vector spaces into a new vector space.

The direct product $\prod _{�\in �}{�}_{�}$ of a family of vector spaces ${�}_{�}$ consists of the set of all tuples ${\displaystyle {\left({\mathbf{�}}_{�}\right)}_{�\in �},}$ which specify for each index $�$ in some index set $�$ an element ${\mathbf{�}}_{�}$ of ${\displaystyle {�}_{�}.}$^[34] Addition and scalar multiplication is performed componentwise. A variant of this construction is the direct sum $\underset{�\in �}{⨁}{�}_{�}$ (also called coproduct and denoted $\coprod _{�\in �}{�}_{�}$), where only tuples with finitely many nonzero vectors are allowed. If the index set $�$ is finite, the two constructions agree, but in general they are different.

Tensor product[edit]

Main article: Tensor product of vector spaces

The tensor product ${\displaystyle �{\otimes }_{�}�,}$ or simply ${\displaystyle �\otimes �,}$ of two vector spaces $�$ and $�$ is one of the central notions of multilinear algebra which deals with extending notions such as linear maps to several variables. A map ${\displaystyle �:�\times �\to �}$ from the Cartesian product $�\times �$ is called bilinear if $�$ is linear in both variables $\mathbf{�}$ and ${\displaystyle \mathbf{�}.}$ That is to say, for fixed $\mathbf{�}$ the map ${\displaystyle \mathbf{�}\mapsto �(\mathbf{�},\mathbf{�})}$ is linear in the sense above and likewise for fixed $\mathbf{�}.$

Commutative diagram depicting the universal property of the tensor product

The tensor product is a particular vector space that is a universal recipient of bilinear maps ${\displaystyle �,}$ as follows. It is defined as the vector space consisting of finite (formal) sums of symbols called tensors

$${\displaystyle {\mathbf{�}}_1\otimes {\mathbf{�}}_1+{\mathbf{�}}_2\otimes {\mathbf{�}}_2+\cdots +{\mathbf{�}}_{�}\otimes {\mathbf{�}}_{�},}$$

subject to the rules^[35]

These rules ensure that the map $�$ from the $�\times �$ to $�\otimes �$ that maps a tuple ${\displaystyle (\mathbf{�},\mathbf{�})}$ to ${\displaystyle \mathbf{�}\otimes \mathbf{�}}$ is bilinear. The universality states that given any vector space $�$ and any bilinear map ${\displaystyle �:�\times �\to �,}$ there exists a unique map $�,$ shown in the diagram with a dotted arrow, whose composition with $�$ equals ${\displaystyle �:}$ ${\displaystyle �(\mathbf{�}\otimes \mathbf{�})=�(\mathbf{�},\mathbf{�}).}$^[36] This is called the universal property of the tensor product, an instance of the method—much used in advanced abstract algebra—to indirectly define objects by specifying maps from or to this object.

Vector spaces with additional structure[edit]

From the point of view of linear algebra, vector spaces are completely understood insofar as any vector space over a given field is characterized, up to isomorphism, by its dimension. However, vector spaces per se do not offer a framework to deal with the question—crucial to analysis—whether a sequence of functions converges to another function. Likewise, linear algebra is not adapted to deal with infinite series, since the addition operation allows only finitely many terms to be added. Therefore, the needs of functional analysis require considering additional structures.

A vector space may be given a partial order ${\displaystyle \le ,}$ under which some vectors can be compared.^[37] For example, $�$-dimensional real space ${\mathbf{�}}^{�}$ can be ordered by comparing its vectors componentwise. Ordered vector spaces, for example Riesz spaces, are fundamental to Lebesgue integration, which relies on the ability to express a function as a difference of two positive functions

$${\displaystyle �={�}^{+}-{�}^{-}.}$$

where ${�}^{+}$ denotes the positive part of $�$ and ${�}^{-}$ the negative part.^[38]