Subspaces of Vector Spaces Math 130 Linear Algebra
... uninteresting subspaces. One is the whole vector space R2 , which is clearly a subspace of itself. A subspace is called a proper subspace if it’s not the entire space, so R2 is the only subspace of R2 which is not a proper subspace. The other obvious and uninteresting subspace is the smallest possib ...
... uninteresting subspaces. One is the whole vector space R2 , which is clearly a subspace of itself. A subspace is called a proper subspace if it’s not the entire space, so R2 is the only subspace of R2 which is not a proper subspace. The other obvious and uninteresting subspace is the smallest possib ...
Appendix B. Vector Spaces Throughout this text we have noted that
... this Appendix when you encounter this concept in the text. B.1. What are vector spaces? In physics and engineering one of the first mathematical concepts that gets introduced is the concept of a vector. Usually, a vector is defined as a quantity that has a direction and a magnitude, such as a positi ...
... this Appendix when you encounter this concept in the text. B.1. What are vector spaces? In physics and engineering one of the first mathematical concepts that gets introduced is the concept of a vector. Usually, a vector is defined as a quantity that has a direction and a magnitude, such as a positi ...
SUBGROUPS OF VECTOR SPACES In what follows, finite
... If G were not closed, there would exist an element g ∈ G \ G and a sequence (gk )k≥1 in G of pairwise distinct elements converging to g. Then (gk − gk+1 )k≥1 is a sequence in G \ {0} converging to zero, contradicting the fact that the origin is isolated in G. ...
... If G were not closed, there would exist an element g ∈ G \ G and a sequence (gk )k≥1 in G of pairwise distinct elements converging to g. Then (gk − gk+1 )k≥1 is a sequence in G \ {0} converging to zero, contradicting the fact that the origin is isolated in G. ...
Here
... that D = ki ∂i , since Df (x) = D(f (a)) + gi (a)D(xi − ai ) = ki ∂i f (a). The upshot of problems 4 and 5 is that the tangent space to Rn can be defined purely algebraically from the ring of smooth functions on Rn . Since this is a local operation, the tangent space of an arbitrary smooth manifold ...
... that D = ki ∂i , since Df (x) = D(f (a)) + gi (a)D(xi − ai ) = ki ∂i f (a). The upshot of problems 4 and 5 is that the tangent space to Rn can be defined purely algebraically from the ring of smooth functions on Rn . Since this is a local operation, the tangent space of an arbitrary smooth manifold ...
Norm and inner products in Rn Math 130 Linear Algebra
... If you draw a triangle with one side being cepts to Rn , but later we’ll abstract these concepts the vector v, and another side the vector w, then to define inner product spaces in general. the third side is w−v. Then the triangle inequality just says that one side of a triangle is less than or The ...
... If you draw a triangle with one side being cepts to Rn , but later we’ll abstract these concepts the vector v, and another side the vector w, then to define inner product spaces in general. the third side is w−v. Then the triangle inequality just says that one side of a triangle is less than or The ...
Chapter 10 Infinite Groups
... The groups we have carefully studied so far are finite groups. In this chapter, we will give a few examples of infinite groups, and revise some of the concepts we have seen in that context. Let us recall a few examples of infinite groups we have seen: • the group of real numbers (with addition), • t ...
... The groups we have carefully studied so far are finite groups. In this chapter, we will give a few examples of infinite groups, and revise some of the concepts we have seen in that context. Let us recall a few examples of infinite groups we have seen: • the group of real numbers (with addition), • t ...
Vector field
In vector calculus, a vector field is an assignment of a vector to each point in a subset of space. A vector field in the plane, for instance, can be visualized as a collection of arrows with a given magnitude and direction each attached to a point in the plane. Vector fields are often used to model, for example, the speed and direction of a moving fluid throughout space, or the strength and direction of some force, such as the magnetic or gravitational force, as it changes from point to point.The elements of differential and integral calculus extend to vector fields in a natural way. When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus. Vector fields can usefully be thought of as representing the velocity of a moving flow in space, and this physical intuition leads to notions such as the divergence (which represents the rate of change of volume of a flow) and curl (which represents the rotation of a flow).In coordinates, a vector field on a domain in n-dimensional Euclidean space can be represented as a vector-valued function that associates an n-tuple of real numbers to each point of the domain. This representation of a vector field depends on the coordinate system, and there is a well-defined transformation law in passing from one coordinate system to the other. Vector fields are often discussed on open subsets of Euclidean space, but also make sense on other subsets such as surfaces, where they associate an arrow tangent to the surface at each point (a tangent vector). More generally, vector fields are defined on differentiable manifolds, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales. In this setting, a vector field gives a tangent vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields are one kind of tensor field.