Linear algebra

 

The three-dimensional Euclidean space R³ is a vector space, and lines and planes passing through the origin are vector subspaces in R³.

Linear algebra is the branch of mathematics concerning finite or countably-infinite dimensional vector spaces, as well as linear mappings between such spaces. Such an investigation is initially motivated by a system of linear equations in several unknowns. Such equations are naturally represented using the formalism of matrices and vectors.

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, 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.

History

The study of linear algebra and matrices first emerged from the study of determinants, which were used to solve systems of linear equations. Determinants were used by Leibniz in 1693, and subsequently, Gabriel Cramer devised Cramer's Rule for solving linear systems in 1750. Later, Gauss further developed the theory of solving linear systems by using Gaussian elimination, which was initially listed as an advancement in geodesy.

The study of matrix algebra first emerged in England in the mid 1800s. In 1848, James Joseph Sylvester introduced the term matrix, which is Latin for "womb". While studying compositions of linear transformations, Arthur Cayley was led to define matrix multiplication and inverses. Crucially, Cayley used a single letter to denote a matrix, thus treating a matrix as an aggregate object. He also realized the connection between matrices and determinants, and wrote "There would be many things to say about this theory of matrices which should, it seems to me, precede the theory of determinants".

The first modern and more precise definition of a vector space was introduced by Peano in 1888; by 1900, a theory of linear transformations of finite-dimensional vector spaces had emerged. Linear algebra first took its modern form in the first half of the twentieth century, when many ideas and methods of previous centuries were generalized as abstract algebra. The use of matrices in quantum mechanics, special relativity, and statistics helped spread the subject of linear algebra beyond pure mathematics. The development of computers led to increased research in efficient algorithms for Gaussian elimination and matrix decompositions, and linear algebra became an essential tool for modelling and simulations.

The origin of many of these ideas is discussed in the articles on determinants and Gaussian elimination.

Recently, Sinologist Roger Hart argued Chinese mathematicians found a method "essentially equivalent to the solution of systems of N equations in N unknowns in modern algebra" a millennium before the West.

Scope of study

Vector spaces

The main structures of linear algebra are vector spaces. A vector space over a field F is a set V together with two binary operations. Elements of V are called vectors and elements of F are called scalars. The first operation, vector addition, takes any two vectors v and w and outputs a third vector v + w. The second operation takes any scalar a and any vector v and outputs a new vector vector av. In view of the first example, where the multiplication is done by rescaling the vector v by a scalar a, the multiplication is called scalar multiplication of v by a. The operations of addition and multiplication in a vector space satisfies the following axioms. In the list below, let u, v and w be arbitrary vectors in V, and a and b scalars in F.

Axiom	Signification
Associativity of addition	u + (v + w) = (u + v) + w
Commutativity of addition	u + v = v + u
Identity element of addition	There exists an element 0 ∈ V, called the zero vector, such that v + 0 = v for all v ∈ V.
Inverse elements of addition	For every v ∈ V, there exists an element −v ∈ V, called the additive inverse of v, such that v + (−v) = 0
Distributivity of scalar multiplication with respect to vector addition	a(u + v) = au + av
Distributivity of scalar multiplication with respect to field addition	(a + b)v = av + bv
Compatibility of scalar multiplication with field multiplication	a(bv) = (ab)v
Identity element of scalar multiplication	1v = v, where 1 denotes the multiplicative identity in F.

Elements of a general vector space V may be objects of any nature, for example, functions, polynomials, vectors, or matrices. Linear algebra is concerned with properties common to all vector spaces.

Linear transformations

Similarly as in the theory of other algebraic structures, linear algebra studies mappings between vector spaces that preserve the vector-space structure. Given two vector spaces V and W over a field F, a linear transformation (also called linear map, linear mapping or linear operator) is a map

$T:V\to W$

that is compatible with addition and scalar multiplication:

$T(u+v)=T(u)+T(v), \quad T(av)=aT(v)$

for any vectors u,v ∈ V and a scalar a ∈ F. When a bijective linear mapping exists between two vector spaces (that is, every vector from the first space is associated with one in the second), we say that the two spaces are isomorphic. Because an isomorphism preserves linear structure, two isomorphic vector spaces are "essentially the same" from the linear algebra point of view. One essential question in linear algebra is whether a mapping is an isomorphism or not, and this question can be answered by checking if the determinant is nonzero. If a mapping is not an isomorphism, linear algebra is interested in finding its range (or image) and the set of elements that get mapped to zero, called the kernel of the mapping.

Subspaces, span, and basis

Again in analogue with theories of other algebraic objects, linear algebra is interested in subsets of vector spaces that are vector spaces themselves; these subsets are called linear subspaces. For instance, the range and kernel of a linear mapping are both subspaces, and are thus often called the range space and the nullspace; these are important examples of subspaces. Another important way of forming a subspace is to take a linear combination of a set of vectors v₁, v₂, …, v_k:

$a_1 v_1 + a_2 v_2 + \cdots + a_k v_k, \,$

where a₁, a₂, …, a_k are scalars. The set of all linear combinations of vectors v₁, v₂, …, v_k is called their span, which forms a subspace.

A linear combination of any system of vectors with all zero coefficients is the zero vector of V. If this is the only way to express zero vector as a linear combination of v₁, v₂, …, v_k then these vectors are linearly independent. Given a set of vectors that span a space, if any vector $w$ is a linear combination of other vectors (and so the set is not linearly independent), then the span would remain the same if we remove $w$ from the set. Thus, a set of linearly dependent vectors is redundant in the sense that a linearly independent subset will span the same subspace. Therefore, we are mostly interested in a linearly independent set of vectors that spans a vector space V, which we call a basis of V. Any set of vectors that spans V contains a basis, and any linearly independent set of vectors in V can be extended to a basis. It turns out that if we accept the axiom of choice, every vector space has a basis; nevertheless, this basis may be unnatural, and indeed, may not even be constructable. For instance, there exists a basis for the real numbers considered as a vector space over the rationals, but no explicit basis has been constructed.

Any two bases of a vector space V have the same cardinality, which is called the dimension of V. The dimension of a vector space is well-defined by the dimension theorem for vector spaces. If a basis of V has finite number of elements, V is called a finite-dimensional vector space. If V is finite-dimensional and U is a subspace of V, then dim U ≤ dim V. If U₁ and U₂ are subspaces of V, then

$\dim(U_1 + U_2) = \dim U_1 + \dim U_2 - \dim(U_1 \cap U_2)$.

One often restricts consideration to finite-dimensional vector spaces. A fundamental theorem of linear algebra states that all vector spaces of the same dimension are isomorphic, giving an easy way of characterizing isomorphism.

Vectors as n-tuples: matrix theory

Main article: Matrix (mathematics)

A particular basis {v₁, v₂, …, v_n} of V allows one to construct a coordinate system in V: the vector with coordinates (a₁, a₂, …, a_n) is the linear combination

$a_1 v_1 + a_2 v_2 + \cdots + a_n v_n. \,$

The condition that v₁, v₂, …, v_n span V guarantees that each vector v can be assigned coordinates, whereas the linear independence of v₁, v₂, …, v_n assures that these coordinates are unique (i.e. there is only one linear combination of the basis vectors that is equal to v). In this way, once a basis of a vector space V over F has been chosen, V may be identified with the coordinate n-space Fⁿ. Under this identification, addition and scalar multiplication of vectors in V correspond to addition and scalar multiplication of their coordinate vectors in Fⁿ. Furthermore, if V and W are an n-dimensional and m-dimensional vector space over F, and a basis of V and a basis of W have been fixed, then any linear transformation T: V → W may be encoded by an m × n matrix A with entries in the field F, called the matrix of T with respect to these bases. Two matrices that encode the same linear transformation in different bases are called similar. Matrix theory replaces the study of linear transformations, which were defined axiomatically, by the study of matrices, which are concrete objects. This major technique distinguishes linear algebra from theories of other algebraic structures, which usually cannot be parametrized so concretely.

There is an important distinction between the coordinate n-space Rⁿ and a general finite-dimensional vector space V. While Rⁿ has a standard basis {e₁, e₂, …, e_n}, a vector space V typically does not come equipped with such a basis and many different bases exist (although they all consist of the same number of elements equal to the dimension of V).

One major application of the matrix theory is calculation of determinants, a central concept in linear algebra. While determinants could be defined in a basis-free manner, they are usually introduced via a specific representation of the mapping; the value of the determinant does not depend on the specific basis. It turns out that a mapping is invertible if and only if the determinant is nonzero. If the determinant is zero, then the nullspace is nontrivial. Determinants have other applications, including a systematic way of seeing if a set of vectors is linearly independent (we write the vectors as the columns of a matrix, and if the determinant of that matrix is zero, the vectors are linearly dependent). Determinants could also be used to solve systems of linear equations (see Cramer's rule), but in real applications, Gaussian elimination is a faster method.

Eigenvalues and eigenvectors

In general, the action of a linear transformation is hard to understand, and so to get a better handle over linear transformations, those vectors that are relatively fixed by that transformation are given special attention. To make this more concrete, let $T: V \to V$ be any linear transformation. We are especially interested in those non-zero vectors $v$ such that $Tv=\lambda v$, where $\lambda$ is a scalar in the base field of the vector space. These vectors are called eigenvectors, and the corresponding scalars are called eigenvalues.

To find an eigenvector or an eigenvalue, we note that

$Tv-\lambda v=(T-\lambda \text{Id})v=0,$

where $\text{Id}$ is the identity matrix. For there to be nontrivial solutions to that equation, $\det(T-\lambda \text{Id})=0$. The determinant is a polynomial, and so the eigenvalues are not guaranteed to exist if the field is R. Thus, we often work with an algebraically closed field such as the complex numbers when dealing with eigenvectors and eigenvalues so that an eigenvalue will always exist. It would be particularly nice if given a transformation $T$ taking a vector space $V$ into itself we can find a basis for $V$ consisting of eigenvectors. If such a basis exists, we can easily compute the action of the transformation on any vector: if $v_1, v_2, \ldots, v_n$ are linearly independent eigenvectors of a mapping of n-dimensional spaces $T$ with (not necessarily distinct) eigenvalues $\lambda_1, \lambda_2, \ldots, \lambda_n$, and if $v=a_1 v_1 + \cdots + a_n v_n$, then,

$T(v)=T(a_1 v_1)+\cdots+T(a_n v_n)=a_1 T(v_1)+\cdots+a_n T(v_n)=a_1 \lambda_1 v_1 + \cdots +a_n \lambda_n v_n.$

Such a transformation is called a diagonalizable matrix since in the eigenbasis, the transformation is represented by a diagonal matrix. Because operations like matrix multiplication, matrix inversion, and determinant calculation are simple on diagonal matrices, computations involving matrices are much simpler if we can bring the matrix to a diagonal form. Not all matrices are diagonalizable (even over an algebraically closed field), but diagonalizable matrices form a dense subset of all matrices.

Inner-product spaces

Besides these basic concepts, linear algebra also studies vector spaces with additional structure, such as an inner product. The inner product is an example of a bilinear form, and it gives the vector space a geometric structure by allowing for the definition of length and angles. Formally, an inner product is a map

$\langle \cdot, \cdot \rangle : V \times V \rightarrow \mathbf{F}$

that satisfies the following three axioms for all vectors $u,v,w \in V$ and all scalars $a \in \mathbf{F}$:

• Conjugate symmetry:

$\langle u,v\rangle =\overline{\langle v,u\rangle}.$

Note that in R, it is symmetric.

• Linearity in the first argument:

$\langle au,v\rangle= a \langle u,v\rangle.$

$\langle u+v,w\rangle= \langle u,w\rangle+ \langle v,w\rangle.$

• Positive-definiteness:

$\langle v,v\rangle \geq 0$ with equality only for $v = 0.$

We can define the length of a vector $v \in V$ by $||v||^2=\langle v,v\rangle$, and we can prove the Cauchy–Schwartz inequality:

$|\langle u,v\rangle| \leq ||u|| \cdot ||v||.$

In particular, the quantity

$\frac{|\langle u,v\rangle|}{||u|| \cdot ||v||} \leq 1,$

and so we can call this quantity the cosine of the angle between the two vectors.

Two vectors are orthogonal if $\langle u, v\rangle =0$. An orthonormal basis is a basis where all basis vectors have length 1 and are orthogonal to each other. Given any finite-dimensional vector space, an orthonormal basis could be found by the Gram–Schmidt procedure. Orthonormal bases are particularly nice to deal with, since if $v=a_1 v_1 + \cdots + a_n v_n$, then $a_i = \langle v,v_i \rangle$.

The inner product facilitates the construction of many useful concepts. For instance, given a transform $T$, we can define its Hermitian conjugate $T^*$ as the linear transform satisfying

$\langle T u, v \rangle = \langle u, T^* v\rangle.$

If T satisfies $T T^*=T^* T$, we call T normal. It turns out that normal matrices are precisely the matrices that have an orthonormal system of eigenvectors that span V.

Some main useful theorems

• A matrix is invertible, or non-singular, if and only if the linear map represented by the matrix is an isomorphism.
• Any vector space over a field F of dimension n is isomorphic to Fⁿ as a vector space over F.
• Corollary: Any two vector spaces over F of the same finite dimension are isomorphic to each other.
• A linear map is an isomorphism if and only if the determinant is nonzero.

Applications

Because of the ubiquity of vector spaces, linear algebra is used in many fields of mathematics, natural sciences, computer science, and social science. Below are just some examples of applications of linear algebra.

Solution of linear systems

Main article: System of linear equations

Linear algebra provides the formal setting for the linear combination of equations used in the Gaussian method. Suppose the goal is to find and describe the solution(s), if any, of the following system of linear equations:

$\begin{alignat}{7} 2x &&\; + \;&& y &&\; - \;&& z &&\; = \;&& 8 & \qquad (L_1) \\ -3x &&\; - \;&& y &&\; + \;&& 2z &&\; = \;&& -11 & \qquad (L_2) \\ -2x &&\; + \;&& y &&\; +\;&& 2z &&\; = \;&& -3 & \qquad (L_3) \end{alignat}$

The Gaussian-elimination algorithm is as follows: eliminate x from all equations below $L_1$, and then eliminate y from all equations below $L_2$. This will put the system into triangular form. Then, using back-substitution, each unknown can be solved for.

In the example, x is eliminated from $L_2$ by adding $\begin{matrix}\frac{3}{2}\end{matrix} L_1$ to $L_2$. x is then eliminated from $L_3$ by adding $L_1$ to $L_3$. Formally:

$L_2 + \frac{3}{2}L_1 \rightarrow L_2$

$L_3 + L_1 \rightarrow L_3$

The result is:

$\begin{alignat}{7} 2x &&\; + && y &&\; - &&\; z &&\; = \;&& 8 & \\ && && \frac{1}{2}y &&\; + &&\; \frac{1}{2}z &&\; = \;&& 1 & \\ && && 2y &&\; + &&\; z &&\; = \;&& 5 & \end{alignat}$

Now y is eliminated from $L_3$ by adding $-4L_2$ to $L_3$:

$L_3 + -4L_2 \rightarrow L_3$

The result is:

$\begin{alignat}{7} 2x &&\; + && y \;&& - &&\; z \;&& = \;&& 8 & \\ && && \frac{1}{2}y \;&& + &&\; \frac{1}{2}z \;&& = \;&& 1 & \\ && && && &&\; -z \;&&\; = \;&& 1 & \end{alignat}$

This result is a system of linear equations in triangular form, and so the first part of the algorithm is complete.

The last part, back-substitution, consists of solving for the knowns in reverse order. It can thus be seen that

$z = -1 \quad (L_3)$

Then, $z$ can be substituted into $L_2$, which can then be solved to obtain

$y = 3 \quad (L_2)$

Next, z and y can be substituted into $L_1$, which can be solved to obtain

$x = 2 \quad (L_1)$

The system is solved.

We can, in general, write any system of linear equations as a matrix equation:

$Ax=b.$

The solution of this system is characterized as follows: first, we find a particular solution $x_0$ of this equation using Gaussian elimination. Then, we compute the solutions of $Ax=0$; that is, we find the nullspace $N$ of A. The solution set of this equation is given by $x_0+N=\{x_0+n: n\in N \}$. If the number of variables equal the number of equations, then we can characterize when the system has a unique solution: since N is trivial if and only if $\det A \neq 0$, the equation has a unique solution if and only if $\det A \neq 0$.

Least-squares best fit line

The least squares method is used to determine the best fit line for a set of data. This line will minimize the sum of the squares of the residuals.

Fourier series expansion

Fourier series are a representation of a function $f:[-\pi,\pi] \to \mathbf{R}$ as a trigonometric series:

$f(x)=\frac{a_0}{2} + \sum_{n=1}^\infty \, [a_n \cos(nx) + b_n \sin(nx)].$

This series expansion is extremely useful in solving partial differential equations. In this article, we will not be concerned with convergence issues; it is nice to note that all continuous functions have a converging Fourier series expansion, and nice enough discontinuous functions have a Fourier series that converges to the function value at most points.

The space of all functions that can be represented by a Fourier series form a vector space (technically speaking, we call functions that have the same Fourier series expansion the "same" function, since two different discontinuous functions might have the same Fourier series). Moreover, this space is also an inner product space with the inner product

$\langle f,g \rangle= \frac{1}{\pi} \int_{-\pi}^\pi f(x) g(x) \, dx.$

The functions $g_n(x)=\sin(nx)$ for $n>0$ and $h_n(x)=\cos(nx)$ for $n \geq 0$ are an orthonormal basis for the space of Fourier-expandable functions. We can thus use the tools of linear algebra to find the expansion of any function in this space in terms of these basis functions. For instance, to find the coefficient $a_k$, we take the inner product with $h_k$:

$\langle f,h_k \rangle=\frac{a_0}{2}\langle h_0,h_k \rangle + \sum_{n=1}^\infty \, [a_n \langle h_n,h_k\rangle + b_n \langle\ g_n,h_k \rangle],$

and by orthonormality, $\langle f,h_k\rangle=a_k$; that is, $a_k = \frac{1}{\pi} \int_{-\pi}^\pi f(x) \cos(kx) \, dx.$

Quantum mechanics

Quantum mechanics is highly inspired by notions in linear algebra. In quantum mechanics, the physical state of a particle is represented by a vector, and observables (such as momentum, energy, and angular momentum) are represented by linear operators on the underlying vector space. More concretely, the wave function of a particle describes its physical state and lies in the vector space L² (the functions $\phi:\mathbf{R}^3 \to \mathbf{C}$ such that $\int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^{\infty} |\phi|^2 \,dx\,dy\,dz$ is finite), and it evolves according to the Schrödinger equation. Energy is represented as the operator $H=-\frac{\hbar^2}{2m} \nabla^2 + V(x,y,z)$, where V is the potential energy. H is also known as the Hamiltonian operator. The eigenvalues of H represents the possible energies that can be observed. Given a particle in some state $\phi$, we can expand $\phi$ into a linear combination of eigenstates of H. The component of H in each eigenstate determines the probability of measuring the corresponding eigenvalue, and the measurement forces the particle to assume that eigenstate (wave function collapse).

Generalizations and related topics

Since linear algebra is a successful theory, its methods have been developed and generalized in other parts of mathematics. In module theory, one replaces the field of scalars by a ring. The concepts of linear independence, span, basis, and dimension (which is called rank in module theory) still make sense. Nevertheless, many theorems from linear algebra become false in module theory. For instance, not all modules have a basis (those that do are called free modules), the rank of a free module is not necessarily unique, not all linearly independent subsets of a module can be extended to form a basis, and not all subsets of a module that span the space contains a basis.

In multilinear algebra, one considers multivariable linear transformations, that is, mappings that are linear in each of a number of different variables. This line of inquiry naturally leads to the idea of the dual space, the vector space $V^*$ consisting of linear maps $f:V \to \mathbf{F}$ where F is the field of scalars. Multilinear maps $T: V^n \to \mathbf{F}$ can be described via tensor products of elements of $V^*$.

If, in addition to vector addition and scalar multiplication, there is a bilinear vector product, then the vector space is called an algebra; for instance, associative algebras are algebras with an associate vector product (like the algebra of square matrices, or the algebra of polynomials).

Functional analysis mixes the methods of linear algebra with those of mathematical analysis and studies various function spaces, such as Lp spaces.

Representation theory studies the actions of algebraic objects on vector spaces by representing these objects as matrices. It is interested in all the ways that this is possible, and it does so by finding subspaces invariant under all transformations of the algebra. The concept of eigenvalues and eigenvectors is especially important.

See also

• List of linear algebra topics
• Numerical linear algebra
• Eigenvectors
• Transformation matrix
• Fundamental matrix in computer vision
• Simplex method, a solution technique for linear programs
• Linear regression, a statistical estimation method

Further reading

History

• Fearnley-Sander, Desmond, "Hermann Grassmann and the Creation of Linear Algebra" ([1]), American Mathematical Monthly 86 (1979), pp. 809–817.
• Grassmann, Hermann, Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik: dargestellt und durch Anwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik, Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert, O. Wigand, Leipzig, 1844.

Introductory textbooks

• Bretscher, Otto (June 28, 2004), Linear Algebra with Applications (3rd ed.), Prentice Hall, ISBN 978-0-13-145334-0 
• Farin, Gerald; Hansford, Dianne (December 15, 2004), Practical Linear Algebra: A Geometry Toolbox, AK Peters, ISBN 978-1-56881-234-2 
• Friedberg, Stephen H.; Insel, Arnold J.; Spence, Lawrence E. (November 11, 2002), Linear Algebra (4th ed.), Prentice Hall, ISBN 978-0-13-008451-4 
• Hefferon, Jim (2008), Linear Algebra, http://joshua.smcvt.edu/linearalgebra/ 
• Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International 
• Lay, David C. (August 22, 2005), Linear Algebra and Its Applications (3rd ed.), Addison Wesley, ISBN 978-0-321-28713-7 
• Kolman, Bernard; Hill, David R. (May 3, 2007), Elementary Linear Algebra with Applications (9th ed.), Prentice Hall, ISBN 978-0-13-229654-0 
• Leon, Steven J. (2006), Linear Algebra With Applications (7th ed.), Pearson Prentice Hall, ISBN 978-0-13-185785-8 
• Poole, David (2010), Linear Algebra: A Modern Introduction (3rd ed.), Cengage – Brooks/Cole, ISBN 978-0-538-73545-2 
• Ricardo, Henry (2010), A Modern Introduction To Linear Algebra (1st ed.), CRC Press, ISBN 978-1-4398-0040-9 

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

Advanced textbooks

• Axler, Sheldon (February 26, 2004), Linear Algebra Done Right (2nd ed.), Springer, ISBN 978-0-387-98258-8 
• Bhatia, Rajendra (November 15, 1996), Matrix Analysis, Graduate Texts in Mathematics, Springer, ISBN 978-0-387-94846-1 
• Demmel, James W. (August 1, 1997), Applied Numerical Linear Algebra, SIAM, ISBN 978-0-89871-389-3 
• Gantmacher, F.R. (2005, 1959 edition), Applications of the Theory of Matrices, Dover Publications, ISBN 978-0-486-44554-0 
• Gantmacher, Felix R. (1990), Matrix Theory Vol. 1 (2nd ed.), American Mathematical Society, ISBN 978-0-8218-1376-8 
• Gantmacher, Felix R. (2000), Matrix Theory Vol. 2 (2nd ed.), American Mathematical Society, ISBN 978-0-8218-2664-5 
• Gelfand, I. M. (1989), Lectures on Linear Algebra, Dover Publications, ISBN 978-0-486-66082-0 
• Glazman, I. M.; Ljubic, Ju. I. (2006), Finite-Dimensional Linear Analysis, Dover Publications, ISBN 978-0-486-45332-3 
• Golan, Johnathan S. (January 2007), The Linear Algebra a Beginning Graduate Student Ought to Know (2nd ed.), Springer, ISBN 978-1-4020-5494-5 
• Golan, Johnathan S. (August 1995), Foundations of Linear Algebra, Kluwer, ISBN 0-7923-3614-3 
• Golub, Gene H.; Van Loan, Charles F. (October 15, 1996), Matrix Computations, Johns Hopkins Studies in Mathematical Sciences (3rd ed.), The Johns Hopkins University Press, ISBN 978-0-8018-5414-9 
• Greub, Werner H. (October 16, 1981), Linear Algebra, Graduate Texts in Mathematics (4th ed.), Springer, ISBN 978-0-8018-5414-9 
• Hoffman, Kenneth; Kunze, Ray (April 25, 1971), Linear Algebra (2nd ed.), Prentice Hall, ISBN 978-0-13-536797-1 
• Halmos, Paul R. (August 20, 1993), Finite-Dimensional Vector Spaces, Undergraduate Texts in Mathematics, Springer, ISBN 978-0-387-90093-3 
• Horn, Roger A.; Johnson, Charles R. (February 23, 1990), Matrix Analysis, Cambridge University Press, ISBN 978-0-521-38632-6 
• Horn, Roger A.; Johnson, Charles R. (June 24, 1994), Topics in Matrix Analysis, Cambridge University Press, ISBN 978-0-521-46713-1 
• Lang, Serge (March 9, 2004), Linear Algebra, Undergraduate Texts in Mathematics (3rd ed.), Springer, ISBN 978-0-387-96412-6 
• Marcus, Marvin; Minc, Henryk (2010), A Survey of Matrix Theory and Matrix Inequalities, Dover Publications, ISBN 978-0-486-67102-4 
• Meyer, Carl D. (February 15, 2001), Matrix Analysis and Applied Linear Algebra, Society for Industrial and Applied Mathematics (SIAM), ISBN 978-0-89871-454-8, http://www.matrixanalysis.com/DownloadChapters.html 
• Mirsky, L. (1990), An Introduction to Linear Algebra, Dover Publications, ISBN 978-0-486-66434-7 
• Roman, Steven (March 22, 2005), Advanced Linear Algebra, Graduate Texts in Mathematics (2nd ed.), Springer, ISBN 978-0-387-24766-3 
• Shilov, Georgi E. (June 1, 1977), Linear algebra, Dover Publications, ISBN 978-0-486-63518-7 
• Shores, Thomas S. (December 6, 2006), Applied Linear Algebra and Matrix Analysis, Undergraduate Texts in Mathematics, Springer, ISBN 978-0-387-33194-2 
• Smith, Larry (May 28, 1998), Linear Algebra, Undergraduate Texts in Mathematics, Springer, ISBN 978-0-387-98455-1 

Study guides and outlines

• Leduc, Steven A. (May 1, 1996), Linear Algebra (Cliffs Quick Review), Cliffs Notes, ISBN 978-0-8220-5331-6 
• Lipschutz, Seymour; Lipson, Marc (December 6, 2000), Schaum's Outline of Linear Algebra (3rd ed.), McGraw-Hill, ISBN 978-0-07-136200-9 
• Lipschutz, Seymour (January 1, 1989), 3,000 Solved Problems in Linear Algebra, McGraw–Hill, ISBN 978-0-07-038023-3 
• McMahon, David (October 28, 2005), Linear Algebra Demystified, McGraw–Hill Professional, ISBN 978-0-07-146579-3 
• Zhang, Fuzhen (April 7, 2009), Linear Algebra: Challenging Problems for Students, The Johns Hopkins University Press, ISBN 978-0-8018-9125-0 

External links

• International Linear Algebra Society
• MIT Professor Gilbert Strang's Linear Algebra Course Homepage : MIT Course Website
• MIT Linear Algebra Lectures: free videos from MIT OpenCourseWare
• Linear Algebra Toolkit.
• Linear Algebra on MathWorld.
• Linear Algebra overview and notation summary on PlanetMath.
• Linear Algebra tutorial with online interactive programs.
• Matrix and Linear Algebra Terms on Earliest Known Uses of Some of the Words of Mathematics
• Earliest Uses of Symbols for Matrices and Vectors on Earliest Uses of Various Mathematical Symbols
• Linear Algebra by Elmer G. Wiens. Interactive web pages for vectors, matrices, linear equations, etc.
• Linear Algebra Solved Problems: Interactive forums for discussion of linear algebra problems, from the lowest up to the hardest level (Putnam).
• Linear Algebra for Informatics. José Figueroa-O'Farrill, University of Edinburgh
• Online Notes / Linear Algebra Paul Dawkins, Lamar University
• Elementary Linear Algebra textbook with solutions
• Linear Algebra Wiki
• Linear algebra (math 21b) homework and exercises
• Textbook and solutions manual, Saylor Foundation.

Online books

• Beezer, Rob, A First Course in Linear Algebra
• Connell, Edwin H., Elements of Abstract and Linear Algebra
• Hefferon, Jim, Linear Algebra
• Matthews, Keith, Elementary Linear Algebra
• Sharipov, Ruslan, Course of linear algebra and multidimensional geometry
• Treil, Sergei, Linear Algebra Done Wrong