Matrix manipulations
... • Not all matrices are square. • Not all matrices are invertible (even nonzero matrices can be singular). • Matrix multiplication is associative, but not commutative. Don’t forget these facts! In matrix computations, we deal not only with the linear algebraic perspective on a matrix, but also with c ...
... • Not all matrices are square. • Not all matrices are invertible (even nonzero matrices can be singular). • Matrix multiplication is associative, but not commutative. Don’t forget these facts! In matrix computations, we deal not only with the linear algebraic perspective on a matrix, but also with c ...
Vector Integral and Differential Calculus (ACM 20150) – Assignment 4
... 1. (a) If u = x3 x̂−xy ŷ +yz 2 ẑ and φ = zx−z 3 y, compute u·∇φ and u×∇φ. Evaluate your results at the point (1, −1, 1). (b) Let u = (yz + 4xy)x̂ + (xz + 2x2 − 3z 2 )ŷ + (xy − 6yz)ẑ. Compute ∇ × u. Find ψ(x, y, z) such that u = ∇ψ. (c) Find the unit outward drawn normal to the surface (x − 2)2 + ...
... 1. (a) If u = x3 x̂−xy ŷ +yz 2 ẑ and φ = zx−z 3 y, compute u·∇φ and u×∇φ. Evaluate your results at the point (1, −1, 1). (b) Let u = (yz + 4xy)x̂ + (xz + 2x2 − 3z 2 )ŷ + (xy − 6yz)ẑ. Compute ∇ × u. Find ψ(x, y, z) such that u = ∇ψ. (c) Find the unit outward drawn normal to the surface (x − 2)2 + ...
![(January 14, 2009) [16.1] Let p be the smallest prime dividing the](http://s1.studyres.com/store/data/001179736_1-17a1d4ec9d3e4b3dafd8254e03147244-300x300.png)
On Some Aspects of the Differential Operator
... THE SPACE L( X, Y ) Let T : X → Y be a linear operator. We denote the collection of all bounded linear operators from X to Y by L(X,Y). The addition and scalar multiplication are introduced to L(X,Y) in a conventional way so that L(X,Y) is a vector space. We mention a theorem, the proof of which fol ...
... THE SPACE L( X, Y ) Let T : X → Y be a linear operator. We denote the collection of all bounded linear operators from X to Y by L(X,Y). The addition and scalar multiplication are introduced to L(X,Y) in a conventional way so that L(X,Y) is a vector space. We mention a theorem, the proof of which fol ...
slides
... is the matrix form of a linear system of equations. We call A the matrix of coefficients, or coefficient matrix ~x the vector of unknowns, or just the unknowns ~b the right-hand-side vector, or just the right-hand-side (rhs). Matlab has a large set of tools useful for solving A~x = ~b. Generically w ...
... is the matrix form of a linear system of equations. We call A the matrix of coefficients, or coefficient matrix ~x the vector of unknowns, or just the unknowns ~b the right-hand-side vector, or just the right-hand-side (rhs). Matlab has a large set of tools useful for solving A~x = ~b. Generically w ...
Solutions to Homework 10
... speed: kv(t)k = 2. Moreover, as t varies from 0 to 4π, a particle completes one revolution. Since 2 × 4π = 8π is the distance traveled over one revolution, this ought to be the circumference of the circle. Is it? Section 10.1 # 8: As in the exercise #6 above, except use t = −1, 0, and 1 for the vect ...
... speed: kv(t)k = 2. Moreover, as t varies from 0 to 4π, a particle completes one revolution. Since 2 × 4π = 8π is the distance traveled over one revolution, this ought to be the circumference of the circle. Is it? Section 10.1 # 8: As in the exercise #6 above, except use t = −1, 0, and 1 for the vect ...
Linear algebra
Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.The set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors.Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations.Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer science, computer animation, and the social sciences (particularly in economics). Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear models.