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Until the 19th century, linear algebra was introduced through systems of linear equations and matrices. In modern mathematics, the presentation through vector spaces is generally preferred, since it is more synthetic, more general not limited to the finite-dimensional case , and conceptually simpler, although more abstract. A vector space over a field F often the field of the real numbers is a set V equipped with two binary operations satisfying the following axioms.

Elements of V are called vectors , and elements of F are called scalars. The second operation, scalar multiplication , takes any scalar a and any vector v and outputs a new vector av. The axioms that addition and scalar multiplication must satisfy are the following. In the list below, u , v and w are arbitrary elements of V , and a and b are arbitrary scalars in the field F. The first four axioms mean that V is an abelian group under addition.

Solving Linear Systems Using Matrices

Elements of a vector space may have various nature; for example, they can be sequences , functions , polynomials or matrices. Linear algebra is concerned with properties common to all vector spaces. Linear maps are mappings between vector spaces that preserve the vector-space structure. Given two vector spaces V and W over a field F , a linear map also called, in some contexts, linear transformation, linear mapping or linear operator is a map.

This implies that for any vectors u , v in V and scalars a , b in F , one has. When a bijective linear map exists between two vector spaces that is, every vector from the second space is associated with exactly one in the first , 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, in the sense that they cannot be distinguished by using vector space properties.

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An essential question in linear algebra is testing whether a linear map is an isomorphism or not, and, if it is not an isomorphism, finding its range or image and the set of elements that are mapped to the zero vector, called the kernel of the map. All these questions can be solved by using Gaussian elimination or some variant of this algorithm. The study of subsets of vector spaces that are themselves vector spaces for the induced operations is fundamental, similarly as for many mathematical structures. These subsets are called linear subspaces.

These conditions suffice for implying that W is a vector space. For example, the image of a linear map, and the inverse image of 0 by a linear map called kernel or null space are linear subspaces.

Another important way of forming a subspace is to consider linear combinations of a set S of vectors: the set of all sums. The span of S is also the intersection of all linear subspaces containing S. In other words, it is the smallest for the inclusion relation linear subspace containing S. A set of vectors is linearly independent if none is in the span of the others. Equivalently, a set S of vector is linearly independent if the only way to express the zero vector as a linear combination of elements of S is to take zero for every coefficient a i.

A set of vectors that spans a vector space is called a spanning set or generating set.

Problems And Solutions In Introductory And Advanced Matrix Calculus

If a spanning set S is linearly dependent that is not linearly independent , then some element w of S is in the span of the other elements of S , and the span would remain the same if one remove w from S. One may continue to remove elements of S until getting a linearly independent spanning set. Such a linearly independent set that spans a vector space V is called a basis of V.

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The importance of bases lies in the fact that there are together minimal generating sets and maximal independent sets. Any two bases of a vector space V have the same cardinality , which is called the dimension of V ; this is the dimension theorem for vector spaces.

Moreover, two vector spaces over the same field F are isomorphic if and only if they have the same dimension. If any basis of V and therefore every basis has a finite number of elements, V is a finite-dimensional vector space. Matrices allow explicit manipulation of finite-dimensional vector spaces and linear maps. Their theory is thus an essential part of linear algebra. Let V be a finite-dimensional vector space over a field F , and v 1 , v 2 , By definition of a basis, the map.

That is, if. Matrix multiplication is defined in such a way that the product of two matrices is the matrix of the composition of the corresponding linear maps, and the product of a matrix and a column matrix is the column matrix representing the result of applying the represented linear map to the represented vector. It follows that the theory of finite-dimensional vector spaces and the theory of matrices are two different languages for expressing exactly the same concepts.

Two matrices that encode the same linear transformation in different bases are called similar. Equivalently, two matrices are similar if one can transform one in the other by elementary row and column operations. For a matrix representing a linear map from W to V , the row operations correspond to change of bases in V and the column operations correspond to change of bases in W.

Every matrix is similar to an identity matrix possibly bordered by zero rows and zero columns. In terms of vector space, this means that, for any linear map from W to V , there are bases such that a part of the basis of W is mapped bijectively on a part of the basis of V , and that the remaining basis elements of W , if any, are mapped to zero this is a way of expressing the fundamental theorem of linear algebra. Gaussian elimination is the basic algorithm for finding these elementary operations, and proving this theorem.

Problems and Solutions in Introductory and Advanced Matrix Calculus : Second Edition -

Systems of linear equations form a fundamental part of linear algebra. Historically, linear algebra and matrix theory has been developed for solving such systems. In the modern presentation of linear algebra through vector spaces and matrices, many problems may be interpreted in terms of linear systems. Let T be the linear transformation associated to the matrix M. A solution of the system S is a vector. Let S' be the associated homogeneous system , where the right-hand sides of the equations are put to zero. The solutions of S' are exactly the elements of the kernel of T or, equivalently, M.

The Gaussian-elimination consists of performing elementary row operations on the augmented matrix. These row operations do not change the set of solutions of the system of equations. In the example, the reduced echelon form is.

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It follows from this matrix interpretation of linear systems that the same methods can be applied for solving linear systems and for many operations on matrices and linear transformations, which include the computation of the ranks , kernels , matrix inverses. A linear endomorphism is a linear map that maps a vector space V to itself. If V has a basis of n elements, such an endomorphism is represented by a square matrix of size n.

With respect to general linear maps, linear endomorphisms and square matrices have some specific properties that make their study an important part of linear algebra, which is used in many parts of mathematics, including geometric transformations , coordinate changes , quadratic forms , and many other part of mathematics. The determinant of a square matrix is a polynomial function of the entries of the matrix, such that the matrix is invertible if and only if the determinant is not zero.

This results from the fact that the determinant of a product of matrices is the product of the determinants, and thus that a matrix is invertible if and only if its determinant is invertible. Cramer's rule is a closed-form expression , in terms of determinants, of the solution of a system of n linear equations in n unknowns.

The determinant of an endomorphism is the determinant of the matrix representing the endomorphism in terms of some ordered basis. This definition makes sense, since this determinant is independent of the choice of the basis. This scalar a is an eigenvalue of f. If the dimension of V is finite, and a basis has been chosen, f and v may be represented, respectively, by a square matrix M and a column matrix z ; the equation defining eigenvectors and eigenvalues becomes.

Using the identity matrix I , whose entries are all zero, except those of the main diagonal, which are equal to one, this may be rewritten. The eigenvalues are thus the roots of the polynomial. If V is of dimension n , this is a monic polynomial of degree n , called the characteristic polynomial of the matrix or of the endomorphism , and there are, at most, n eigenvalues. If a basis exists that consists only of eigenvectors, the matrix of f on this basis has a very simple structure: it is a diagonal matrix such that the entries on the main diagonal are eigenvalues, and the other entries are zero.

In this case, the endomorphism and the matrix are said diagonalizable. More generally, an endomorphism and a matrix are also said diagonalizable, if they become diagonalizable after extending the field of scalars. In this extended sense, if the characteristic polynomial is square-free , then the matrix is diagonalizable.

A symmetric matrix is always diagonalizable. There are non-diagonalizable matrices, the simplest being. When an endomorphism is not diagonalizable, there are bases on which it has a simple form, although not as simple as the diagonal form. The Frobenius normal form does not need of extending the field of scalars and makes the characteristic polynomial immediately readable on the matrix.

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