1 Equivalence Relations
... Let V be a vector space over the field k and let U be a subspace of V . From this data, we will construct a new vector space V /U called the quotient space whose vectors are equivalence classes of vectors from V and whose operations of addition and scalar multiplication are induced by the correspond ...
... Let V be a vector space over the field k and let U be a subspace of V . From this data, we will construct a new vector space V /U called the quotient space whose vectors are equivalence classes of vectors from V and whose operations of addition and scalar multiplication are induced by the correspond ...
Math 215 HW #4 Solutions
... Solution: If A is an m × n matrix and Ax = b always has at least one solution for any choice of b ∈ Rm , that means that any vector b ∈ Rm must lie in the column space of A. In other words, col(A) = Rm . This, then, means that the dimensions of these two spaces must be the same: dim col(A) = dim Rm ...
... Solution: If A is an m × n matrix and Ax = b always has at least one solution for any choice of b ∈ Rm , that means that any vector b ∈ Rm must lie in the column space of A. In other words, col(A) = Rm . This, then, means that the dimensions of these two spaces must be the same: dim col(A) = dim Rm ...
here
... This sequence is not geometric, because from the first two terms we would have to have x1 = 2, k = 3/4, but then this would require the third term to be k 2 x1 = 9/8, which it is not. Therefore, the set of geometric sequences is not closed under addition, so it is not a subspace of R∞ . 2. Explanati ...
... This sequence is not geometric, because from the first two terms we would have to have x1 = 2, k = 3/4, but then this would require the third term to be k 2 x1 = 9/8, which it is not. Therefore, the set of geometric sequences is not closed under addition, so it is not a subspace of R∞ . 2. Explanati ...
10 The Singular Value Decomposition
... Figure 33: Decomposition of the mapping in figure 32. The singular value decomposition is “almost unique”. There are two sources of ambiguity. The first is in the orientation of the singular vectors. One can flip any right singular vector, provided that the corresponding left singular vector is flip ...
... Figure 33: Decomposition of the mapping in figure 32. The singular value decomposition is “almost unique”. There are two sources of ambiguity. The first is in the orientation of the singular vectors. One can flip any right singular vector, provided that the corresponding left singular vector is flip ...
Definitions of Linear Algebra Terms
... 1. Carefully read a section in the textbook and also read your class notes. Pay attention to de…nitions of terms and examples and try to understand each concept along the way as you read it. You don’t have to memorize all of the de…nitions upon …rst reading. You will memorize them gradually as you w ...
... 1. Carefully read a section in the textbook and also read your class notes. Pay attention to de…nitions of terms and examples and try to understand each concept along the way as you read it. You don’t have to memorize all of the de…nitions upon …rst reading. You will memorize them gradually as you w ...
Coding Theory: Homework 1
... Suppose we have an [n, k, d]2m with full rank G generating matrix of size k × n, which exists by Theorem 2.2.1. Create a new matrix G0 of size km × nm by the following procedure: For each a(i, j) in G will be replaced by an m × m block matrix. The first row will be the representation of a(i, j) unde ...
... Suppose we have an [n, k, d]2m with full rank G generating matrix of size k × n, which exists by Theorem 2.2.1. Create a new matrix G0 of size km × nm by the following procedure: For each a(i, j) in G will be replaced by an m × m block matrix. The first row will be the representation of a(i, j) unde ...
Math 133, Chapter 11 Practice 1. The cranes in the following figure
... This is the parallelogram identity, because a parallelogram whose sides are parallel to u and v has diagonals u + v and u − v. Thus the sum of the squares of the lengths of the four sides is equal to the sum of the squares of the lengths of the diagonals. 13. (a) Do the points (2, 9, 1), (3, 11, 4), ...
... This is the parallelogram identity, because a parallelogram whose sides are parallel to u and v has diagonals u + v and u − v. Thus the sum of the squares of the lengths of the four sides is equal to the sum of the squares of the lengths of the diagonals. 13. (a) Do the points (2, 9, 1), (3, 11, 4), ...
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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.