Physics 3730/6720 – Maple 1b – 1 Linear algebra, Eigenvalues and Eigenvectors
... their elements. To do matrix and vector products and see the result you need evalm. (Maple does it, otherwise, but silently.) Multiplication by scalars works with *, but multiplication of matrices with matrices and matrices with vectors requires the special multiplication operator &*. > eigenvals(A) ...
... their elements. To do matrix and vector products and see the result you need evalm. (Maple does it, otherwise, but silently.) Multiplication by scalars works with *, but multiplication of matrices with matrices and matrices with vectors requires the special multiplication operator &*. > eigenvals(A) ...
linear combination
... The R stands for the real numbers that appear as entries in the vector, and the exponent 2 indicates that each vector contains 2 entries. Two vectors in R2 are equal if and only if their corresponding entries are equal. Given two vectors u and v in R2, their sum is the vector u v obtained by ...
... The R stands for the real numbers that appear as entries in the vector, and the exponent 2 indicates that each vector contains 2 entries. Two vectors in R2 are equal if and only if their corresponding entries are equal. Given two vectors u and v in R2, their sum is the vector u v obtained by ...
Solutions
... All three of the methods for Problem 1 share this feature, namely that there are two valid loop orderings for each of the methods. It should be noted that the different loop orderings do not correspond to different methods of solving the problem, though, just different ways of expressing the mathema ...
... All three of the methods for Problem 1 share this feature, namely that there are two valid loop orderings for each of the methods. It should be noted that the different loop orderings do not correspond to different methods of solving the problem, though, just different ways of expressing the mathema ...
Math 480 Notes on Orthogonality The word orthogonal is a synonym
... We now consider in detail the question of why every subspace of Rn has a basis. Theorem 3. If S is a subspace of Rn , then S has a basis containing at most n elements. Equivalently, dim(S) 6 n. Proof. First, recall that every set of n + 1 (or more) vectors in Rn is linearly dependent, since they for ...
... We now consider in detail the question of why every subspace of Rn has a basis. Theorem 3. If S is a subspace of Rn , then S has a basis containing at most n elements. Equivalently, dim(S) 6 n. Proof. First, recall that every set of n + 1 (or more) vectors in Rn is linearly dependent, since they for ...
![1. Let A = 1 −1 1 1 0 −1 2 1 1 . a) [2 marks] Find the](http://s1.studyres.com/store/data/005284378_1-9abef9398f6a7d24059a09f56fe1ac13-300x300.png)
PDF
... The proof of this assertion is straightforward. Each of the brackets in the lefthand side expands to 4 terms, and then everything cancels. In categorical terms, what we have here is a functor from the category of associative algebras to the category of Lie algebras over a fixed field. The action of ...
... The proof of this assertion is straightforward. Each of the brackets in the lefthand side expands to 4 terms, and then everything cancels. In categorical terms, what we have here is a functor from the category of associative algebras to the category of Lie algebras over a fixed field. The action of ...
Vector space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied (""scaled"") by numbers, called scalars in this context. Scalars are often taken to be real numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers, or generally any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called axioms, listed below. Euclidean vectors are an example of a vector space. They represent physical quantities such as forces: any two forces (of the same type) can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, vectors representing displacements in the plane or in three-dimensional space also form vector spaces. Vectors in vector spaces do not necessarily have to be arrow-like objects as they appear in the mentioned examples: vectors are regarded as abstract mathematical objects with particular properties, which in some cases can be visualized as arrows.Vector spaces are the subject of linear algebra and are well understood from this point of view since vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. A vector space may be endowed with additional structure, such as a norm or inner product. Such spaces arise naturally in mathematical analysis, mainly in the guise of infinite-dimensional function spaces whose vectors are functions. Analytical problems call for the ability to decide whether a sequence of vectors converges to a given vector. This is accomplished by considering vector spaces with additional structure, mostly spaces endowed with a suitable topology, thus allowing the consideration of proximity and continuity issues. These topological vector spaces, in particular Banach spaces and Hilbert spaces, have a richer theory.Historically, the first ideas leading to vector spaces can be traced back as far as the 17th century's analytic geometry, matrices, systems of linear equations, and Euclidean vectors. The modern, more abstract treatment, first formulated by Giuseppe Peano in 1888, encompasses more general objects than Euclidean space, but much of the theory can be seen as an extension of classical geometric ideas like lines, planes and their higher-dimensional analogs.Today, vector spaces are applied throughout mathematics, science and engineering. They are the appropriate linear-algebraic notion to deal with systems of linear equations; offer a framework for Fourier expansion, which is employed in image compression routines; or provide an environment that can be used for solution techniques for partial differential equations. Furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques. Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and abstract algebra.