Exam No. 01 (Fall 2013) PHYS 320: Electricity and Magnetism I
... f (x, y, z) = cos x sin y + cos y sin z + cos z sin x ...
... f (x, y, z) = cos x sin y + cos y sin z + cos z sin x ...
Lesson 10 - Linear and Nonlinear Functions
... Example 6 Identify Functions Using Tables Determine whether the table represents a linear or nonlinear function. Explain. As x increases by 1, y increases by a greater amount each time. The rate of change is not constant, so this function is nonlinear. ...
... Example 6 Identify Functions Using Tables Determine whether the table represents a linear or nonlinear function. Explain. As x increases by 1, y increases by a greater amount each time. The rate of change is not constant, so this function is nonlinear. ...
Basis and Dimension The Dimension Theorem Every basis of V has
... and that y1 , y2 , . . . , yd are vectors in V . Then the following are equivalent. a) {y1 , y2 , . . . , yd } is a basis of V . b) {y1 , y2 , . . . , yd } are linearly independent. c) {y1 , y2 , . . . , yd } span V . ...
... and that y1 , y2 , . . . , yd are vectors in V . Then the following are equivalent. a) {y1 , y2 , . . . , yd } is a basis of V . b) {y1 , y2 , . . . , yd } are linearly independent. c) {y1 , y2 , . . . , yd } span V . ...
test one
... 21. Given the line y = 4x + 3 (a) state the line’s slope (b) state the slope of a line which is perpendicular to y = 4x + 3 (c) find an equation for the line which is perpendicular to y = 4x + 3 and which passes through the point (−8, 5). 22. Suppose there is a linear relation between the distance t ...
... 21. Given the line y = 4x + 3 (a) state the line’s slope (b) state the slope of a line which is perpendicular to y = 4x + 3 (c) find an equation for the line which is perpendicular to y = 4x + 3 and which passes through the point (−8, 5). 22. Suppose there is a linear relation between the distance t ...
Problem Set 2
... is in the group SL(2, C), but is not of the form eX for any X ∈ sl(2, C) (this Lie algebra is all 2 by 2 matrices with trace zero. Hint: For 2 by 2 matrices X, one can show (this is the Cayley-Hamilton theorem: matrices X satisfy their own characteristic equation det(λ1 − X) = 0, and for 2 by 2 mat ...
... is in the group SL(2, C), but is not of the form eX for any X ∈ sl(2, C) (this Lie algebra is all 2 by 2 matrices with trace zero. Hint: For 2 by 2 matrices X, one can show (this is the Cayley-Hamilton theorem: matrices X satisfy their own characteristic equation det(λ1 − X) = 0, and for 2 by 2 mat ...
1-3 Transforming Linear functions
... • The student will be able to: • Transform linear functions • Solve problems involving linear transformations ...
... • The student will be able to: • Transform linear functions • Solve problems involving linear transformations ...
Algebra Brain Summary
... Determinants A special operation performed on a matrix Properties of Determinants (jump) – A list of properties that we can use to simplify the process of finding determinants Applications of Determinants Several useful ways to apply matrices Area of Triangle How to find the area of a triangle ...
... Determinants A special operation performed on a matrix Properties of Determinants (jump) – A list of properties that we can use to simplify the process of finding determinants Applications of Determinants Several useful ways to apply matrices Area of Triangle How to find the area of a triangle ...
(pdf)
... Proof. By the definition of ε and µ we have (ε ◦ µ)(xy, z) = ε(xyz) = (ε ◦ µ)(x, yz) for all x, y, z ∈ A. Since ε is linear, we also have (ε ◦ µ)(x1 + x2 , y) = ε((x1 + x2 )y) = ε(x1 y + x2 y) = ε(x1 y) + ε(x2 y) = (ε ◦ µ)(x1 , y) + (ε ◦ µ)(x2 , y) and (ε ◦ µ)(ax, y) = ε(axy) = aε(xy) = a(ε ◦ µ)(x, ...
... Proof. By the definition of ε and µ we have (ε ◦ µ)(xy, z) = ε(xyz) = (ε ◦ µ)(x, yz) for all x, y, z ∈ A. Since ε is linear, we also have (ε ◦ µ)(x1 + x2 , y) = ε((x1 + x2 )y) = ε(x1 y + x2 y) = ε(x1 y) + ε(x2 y) = (ε ◦ µ)(x1 , y) + (ε ◦ µ)(x2 , y) and (ε ◦ µ)(ax, y) = ε(axy) = aε(xy) = a(ε ◦ µ)(x, ...
Math Summer Camp Weekly Schedule, Content, and Reading/Video
... A set can be thought of as a collection of distinct things united by some common feature. Mathematical logic, also known as symbolic logic, was developed when people finally realized that the tools of mathematics can be used to study the structure of logic itself. Geometry and Topology Geometry dea ...
... A set can be thought of as a collection of distinct things united by some common feature. Mathematical logic, also known as symbolic logic, was developed when people finally realized that the tools of mathematics can be used to study the structure of logic itself. Geometry and Topology Geometry dea ...
Linear dependence and independence (chapter. 4)
... Suppose that V is a vector space. The set of vectors {x1 , x2 , . . . , xk } in V is linearly independent if the only scalars r1 , r2 , . . . , rk ∈ R such that r1 x1 + r2 x2 + · · · + rk xk = 0 are r1 = r2 = · · · = rk = 0. (That is, {x1 , . . . , xk } is not linearly dependent!) • If {x1 , x2 , . ...
... Suppose that V is a vector space. The set of vectors {x1 , x2 , . . . , xk } in V is linearly independent if the only scalars r1 , r2 , . . . , rk ∈ R such that r1 x1 + r2 x2 + · · · + rk xk = 0 are r1 = r2 = · · · = rk = 0. (That is, {x1 , . . . , xk } is not linearly dependent!) • If {x1 , x2 , . ...
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.