Math39104-Notes - Department of Mathematics, CCNY
... are not. Instead, starting with two functions f and g the right hand side in each case is used to define the new functions labeled f + g and af . Given a finite list of vectors, like v1 , v2 , v3 , it is possible to build from them infinitely many vectors from them by using the vector space operations. ...
... are not. Instead, starting with two functions f and g the right hand side in each case is used to define the new functions labeled f + g and af . Given a finite list of vectors, like v1 , v2 , v3 , it is possible to build from them infinitely many vectors from them by using the vector space operations. ...
Homework 8
... (5) An n × n matrix A = (aij ) is upper triangular if all the entries below the diagonal are zero, i.e. if aij = 0 for i > j. Show that, if A and B are upper-triangular n × n matrices, then so is AB. (6) (*)Let V = R[X]3 , the vector space of polynomials of degree 3, and let B be the ordered basis ( ...
... (5) An n × n matrix A = (aij ) is upper triangular if all the entries below the diagonal are zero, i.e. if aij = 0 for i > j. Show that, if A and B are upper-triangular n × n matrices, then so is AB. (6) (*)Let V = R[X]3 , the vector space of polynomials of degree 3, and let B be the ordered basis ( ...
Lectures five and six
... (respectively Lie algebra) is said to be irreducible(defined) if the only invariant subspaces for the representation are the trivial space and the whole space. In other words, there are no proper nontrivial invariant subspaces. ...
... (respectively Lie algebra) is said to be irreducible(defined) if the only invariant subspaces for the representation are the trivial space and the whole space. In other words, there are no proper nontrivial invariant subspaces. ...
6.1 Change of Basis
... One way the formula v · w |v| |w| cos θ is useful is that it allows us to find the angle between two vectors. To use it in this fashion, we solve for the angle θ: Because of the inverse cosine, this formula always yields an angle between 0◦ and 180◦ . ...
... One way the formula v · w |v| |w| cos θ is useful is that it allows us to find the angle between two vectors. To use it in this fashion, we solve for the angle θ: Because of the inverse cosine, this formula always yields an angle between 0◦ and 180◦ . ...
Chapter 1 Notes
... c) The volume of the soft drink can is 0.360 liters. d) The velocity of the rocket was 325 m/s, due east. e) The light took approximately 500 s to travel from the sun to the earth. ...
... c) The volume of the soft drink can is 0.360 liters. d) The velocity of the rocket was 325 m/s, due east. e) The light took approximately 500 s to travel from the sun to the earth. ...
PDF
... Since i = "1 and i 2 = i " i = #1, it is consistent to represent multiplying a real number by i as the rotation of a point on the real number axis 90º counterclockwise about the origin to a point equidistant from the origin on the imaginary axis. ...
... Since i = "1 and i 2 = i " i = #1, it is consistent to represent multiplying a real number by i as the rotation of a point on the real number axis 90º counterclockwise about the origin to a point equidistant from the origin on the imaginary axis. ...
Common Core State Standards for Mathematics -
... Explain why the x-coordinate of the points where the graphs of the equations y=f(x) and y=g(x) intersect are solutions of the equations f(x)=g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases ...
... Explain why the x-coordinate of the points where the graphs of the equations y=f(x) and y=g(x) intersect are solutions of the equations f(x)=g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases ...
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.