linear algebra - Universitatea "Politehnica"
... Examples of vector spaces (i) K as a vector space over itself, with addition and multiplication in K. (ii) Kn is a K–vector space. (iii) Mm,n (K) with usual matrix addition and multiplication by scalars is a K–vector space. (iv) The set of space vectors V3 is a real vector space with addition given ...
... Examples of vector spaces (i) K as a vector space over itself, with addition and multiplication in K. (ii) Kn is a K–vector space. (iii) Mm,n (K) with usual matrix addition and multiplication by scalars is a K–vector space. (iv) The set of space vectors V3 is a real vector space with addition given ...
Phase-space invariants for aggregates of particles: Hyperangular
... presented of the previous work43 which had treated the case of N⬎4 and d⫽3). By this approach, the singular values are simply related to the principal moments of inertia of the system 共accordingly they are often referred to as principal axis coordinates兲, while the remaining coordinates are related ...
... presented of the previous work43 which had treated the case of N⬎4 and d⫽3). By this approach, the singular values are simply related to the principal moments of inertia of the system 共accordingly they are often referred to as principal axis coordinates兲, while the remaining coordinates are related ...
Mathematics for Economic Analysis I
... If there are two sets A and B and for each element of A, a unique element of B is assigned in some manner or other we have a function or a functional relation from the set A to the set B. Such an assignment, which is a special kind of relation, is usually written as ...
... If there are two sets A and B and for each element of A, a unique element of B is assigned in some manner or other we have a function or a functional relation from the set A to the set B. Such an assignment, which is a special kind of relation, is usually written as ...
Chapter 2 Defn 1. - URI Math Department
... Defn 2. (p. 67) Let V and W be vector spaces, and let T : V → W be linear. We define the null space (or kernel) N (T ) of T to be the set of all vectors x in V such that T (x) = 0; N (T ) = {x ∈ V : T (x) = 0}. We define the range (or image) R(T ) of T to be the subset W consisting of all images (un ...
... Defn 2. (p. 67) Let V and W be vector spaces, and let T : V → W be linear. We define the null space (or kernel) N (T ) of T to be the set of all vectors x in V such that T (x) = 0; N (T ) = {x ∈ V : T (x) = 0}. We define the range (or image) R(T ) of T to be the subset W consisting of all images (un ...
Linear Equations - Number Theory Web
... has only the trivial solution, whereas the homogeneous system ...
... has only the trivial solution, whereas the homogeneous system ...
Solution for Linear Systems
... Elementary Row Operations on a Matrix There are three elementary row operations on a matrix. They are Interchange of any two Rows. Multiplication of the elements of any row with a non-zero scalar (or constant) Multiplication of elements of a row with a scalar and added to the corresponding elements ...
... Elementary Row Operations on a Matrix There are three elementary row operations on a matrix. They are Interchange of any two Rows. Multiplication of the elements of any row with a non-zero scalar (or constant) Multiplication of elements of a row with a scalar and added to the corresponding elements ...
isometric immersions of lorentz space with parallel second
... Let x(s) be a non-null curve in an indefinite euclidean space ...
... Let x(s) be a non-null curve in an indefinite euclidean space ...
Jordan normal form
In linear algebra, a Jordan normal form (often called Jordan canonical form)of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing the operator with respect to some basis. Such matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them. If the vector space is over a field K, then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is the field of complex numbers. The diagonal entries of the normal form are the eigenvalues of the operator, with the number of times each one occurs being given by its algebraic multiplicity.If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size.The Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form.The Jordan normal form is named after Camille Jordan.