Homogeneous equations, Linear independence
... Other solutions called nontrivial solutions. Theorem 1: A nontrivial solution of Ð$Ñ exists iff [if and only if] the system has at least one free variable in row echelon form. The same is true for any homogeneous system of equations. Proof: If there are no free variables, there is only one solution ...
... Other solutions called nontrivial solutions. Theorem 1: A nontrivial solution of Ð$Ñ exists iff [if and only if] the system has at least one free variable in row echelon form. The same is true for any homogeneous system of equations. Proof: If there are no free variables, there is only one solution ...
Problem Set #7 Solutions
... Proof. To prove that W is a subspace, we need to show three things: (1) The set W is not the empty set. In other words, that W contains at least one element. (2) The set W is closed under addition. In other words, that given any two elements v, u ∈ W , their sum v + u is also an element of W . (3) T ...
... Proof. To prove that W is a subspace, we need to show three things: (1) The set W is not the empty set. In other words, that W contains at least one element. (2) The set W is closed under addition. In other words, that given any two elements v, u ∈ W , their sum v + u is also an element of W . (3) T ...
Linear operators whose domain is locally convex
... affinely homeomorphic to a subset of a locally convex space. This is immediate, since the topology on T(U) can be induced by the family of affine functionals vanishing at 0. It is also equivalent to the condition that 0 has a base of convex neighbourhoods in T(U); this is proved by constructing on t ...
... affinely homeomorphic to a subset of a locally convex space. This is immediate, since the topology on T(U) can be induced by the family of affine functionals vanishing at 0. It is also equivalent to the condition that 0 has a base of convex neighbourhoods in T(U); this is proved by constructing on t ...
Lecture 1 - Lie Groups and the Maurer-Cartan equation
... Let θ be a the g-valued 1-form that assigns to a vector v ∈ TA G is associated element of g. In some sense, θ is an identity automorphism. It is common to write θA = A−1 dA. ...
... Let θ be a the g-valued 1-form that assigns to a vector v ∈ TA G is associated element of g. In some sense, θ is an identity automorphism. It is common to write θA = A−1 dA. ...
Lecture 1 Linear Superalgebra
... The category of super vector spaces admits tensor products: Given the super vector spaces V, W , V ⊗ W has the following natural Z/2Z-grading: (V ⊗ W )0 = (V0 ⊗ W0 ) ⊕ (V1 ⊗ W1 ) (V ⊗ W )1 = (V0 ⊗ W1 ) ⊕ (V1 ⊗ W0 ). The assignment V, W 7−→ V ⊗ W is additive and exact in each variable as in the ordin ...
... The category of super vector spaces admits tensor products: Given the super vector spaces V, W , V ⊗ W has the following natural Z/2Z-grading: (V ⊗ W )0 = (V0 ⊗ W0 ) ⊕ (V1 ⊗ W1 ) (V ⊗ W )1 = (V0 ⊗ W1 ) ⊕ (V1 ⊗ W0 ). The assignment V, W 7−→ V ⊗ W is additive and exact in each variable as in the ordin ...
Example 1: Velocity, Acceleration and Speed.
... As an object moves along a curve in the plane, the coordinates x and y of its center of mass are each functions of time t. Rather than use f and g to represent two functions, we write x x(t ) and y y (t ) . So our position vector r (t ) is in the form r (t ) x(t )i y (t ) j . The nice thing ...
... As an object moves along a curve in the plane, the coordinates x and y of its center of mass are each functions of time t. Rather than use f and g to represent two functions, we write x x(t ) and y y (t ) . So our position vector r (t ) is in the form r (t ) x(t )i y (t ) j . The nice thing ...
Sept. 3, 2013 Math 3312 sec 003 Fall 2013
... Let T : Rn −→ Rm be a linear transformation. There exists a unique m × n matrix A such that T (x) = Ax for every x ∈ Rn . Moreover, the j th column of the matrix A is the vector T (ej ), where ej is the j th column of the n × n identity matrix In . That is, A = [T (e1 ) T (e2 ) ...
... Let T : Rn −→ Rm be a linear transformation. There exists a unique m × n matrix A such that T (x) = Ax for every x ∈ Rn . Moreover, the j th column of the matrix A is the vector T (ej ), where ej is the j th column of the n × n identity matrix In . That is, A = [T (e1 ) T (e2 ) ...
LINEAR COMBINATIONS AND SUBSPACES
... have u = (1, 0) ∈ S and v = (0, 1) ∈ S but u + v = (1, 1) does not belong to S since 12 + 12 = 2 6≤ 1. Properties of Subspaces (1) Every subspace of Rn contains the zero vector. (2) If U is a nonempty finite subset of Rn then the span of U is a subspace of Rn and is called the subspace spanned or ge ...
... have u = (1, 0) ∈ S and v = (0, 1) ∈ S but u + v = (1, 1) does not belong to S since 12 + 12 = 2 6≤ 1. Properties of Subspaces (1) Every subspace of Rn contains the zero vector. (2) If U is a nonempty finite subset of Rn then the span of U is a subspace of Rn and is called the subspace spanned or ge ...
Quotient spaces defined by linear relations
... and thus / is continuous. Moreover, by Corollary 3.3, S is lower semi-continuous. Finally, if U is an open subset of X, then by Theorem 1.4, S(U) = (р~^((рз{^У), whence we can infer that S(U) is open in Y, since (ps is open and (p is continuous. This proves that S is open. Corollary 3.7. Let S be a ...
... and thus / is continuous. Moreover, by Corollary 3.3, S is lower semi-continuous. Finally, if U is an open subset of X, then by Theorem 1.4, S(U) = (р~^((рз{^У), whence we can infer that S(U) is open in Y, since (ps is open and (p is continuous. This proves that S is open. Corollary 3.7. Let S be a ...
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.