Differential Equations with Linear Algebra

... 3.1. Vector Spaces and Dimension. (10 points) For each of the following spaces, show whether or not it is a vector space over the scalar field R. If it is a vector space, give its dimension. (a) Symmetric 2 × 2 real matrices, i.e. matrices A such that the transpose AT is equal to A (with respect to ...

... 3.1. Vector Spaces and Dimension. (10 points) For each of the following spaces, show whether or not it is a vector space over the scalar field R. If it is a vector space, give its dimension. (a) Symmetric 2 × 2 real matrices, i.e. matrices A such that the transpose AT is equal to A (with respect to ...

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... determinant of A2 (b) to also be 5, therefore, applying Cramer’s rule, we see that x2 = 55 = 1. (4) (a) This is NOT a vector space, consider the zero vector in this space–it must be the zero polynomial, and the zero polynomial is NOT of the form a + t2 for any real number a. This also fails to be cl ...

... determinant of A2 (b) to also be 5, therefore, applying Cramer’s rule, we see that x2 = 55 = 1. (4) (a) This is NOT a vector space, consider the zero vector in this space–it must be the zero polynomial, and the zero polynomial is NOT of the form a + t2 for any real number a. This also fails to be cl ...

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... is a sequence of topological vector spaces and d = (dq )q∈Z is a sequence of continuous linear maps dq from Eq+1 into Eq which satisfy dq ◦ dq+1 = 0. Remarks • The homological complex of topological vector spaces is a specifc example of a chain complex. • A sequence of R-modules and their homomorphi ...

... is a sequence of topological vector spaces and d = (dq )q∈Z is a sequence of continuous linear maps dq from Eq+1 into Eq which satisfy dq ◦ dq+1 = 0. Remarks • The homological complex of topological vector spaces is a specifc example of a chain complex. • A sequence of R-modules and their homomorphi ...

Sample Exam 2

... Show your work in the spaces provided below for full credit. Use the reverse side for additional space, but clearly so indicate. You must clearly identify answers and show supporting work to receive any credit. Exact answers (e.g., π ) are preferred to inexact (e.g., 3.14). Point values of problems ...

... Show your work in the spaces provided below for full credit. Use the reverse side for additional space, but clearly so indicate. You must clearly identify answers and show supporting work to receive any credit. Exact answers (e.g., π ) are preferred to inexact (e.g., 3.14). Point values of problems ...

ex2m314smp.pdf

... Show your work in the spaces provided below for full credit. Use the reverse side for additional space, but clearly so indicate. You must clearly identify answers and show supporting work to receive any credit. Exact answers (e.g., π ) are preferred to inexact (e.g., 3.14). Point values of problems ...

... Show your work in the spaces provided below for full credit. Use the reverse side for additional space, but clearly so indicate. You must clearly identify answers and show supporting work to receive any credit. Exact answers (e.g., π ) are preferred to inexact (e.g., 3.14). Point values of problems ...

Homework 6

... (5) Show that R is a vector space over Q, with vector addition given by addition in R √ and scalar multiplication given by multiplication in R. Show that {1, 2} are linearly independent vectors in the Q-vector space R. (6) (*) Let V be the vector space over F of sequences (an )n∈N of elements of F . ...

... (5) Show that R is a vector space over Q, with vector addition given by addition in R √ and scalar multiplication given by multiplication in R. Show that {1, 2} are linearly independent vectors in the Q-vector space R. (6) (*) Let V be the vector space over F of sequences (an )n∈N of elements of F . ...

LINEABILITY WITHIN PROBABILITY THEORY SETTINGS 1

... cial or unexpected properties. Vector spaces and linear algebras are elegant ...

... cial or unexpected properties. Vector spaces and linear algebras are elegant ...

EE 550 Lecture no. 9

... α (βx) = (αβ) x (Scalar multiplication is associative) 8. 1x = x where 1 is the multiplicative identity in 9. For any α in and any x1 and x2 in X α (x1 + x2) = α x1 + α x2 (scalar multiplication is distributive w.r.t. vector addition) 10. For any α, β in and all x in X (α + β) x = α x + β x (S ...

... α (βx) = (αβ) x (Scalar multiplication is associative) 8. 1x = x where 1 is the multiplicative identity in 9. For any α in and any x1 and x2 in X α (x1 + x2) = α x1 + α x2 (scalar multiplication is distributive w.r.t. vector addition) 10. For any α, β in and all x in X (α + β) x = α x + β x (S ...

TRIANGLES: 1. Law of Sines

... NOTE: Don’t forget about the Ambiguous Case (SSA or ASS) 2. Law of Cosines (use this when you have SAS or SSS) ...

... NOTE: Don’t forget about the Ambiguous Case (SSA or ASS) 2. Law of Cosines (use this when you have SAS or SSS) ...

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.