Linear Systems of Differential Equations

Linear Systems of Di erential Equations Math 240 First order linear systems Solutions Beyond rst order systems Solutions to homogeneous linear systems As with linear systems, a homogeneous linear system of di erential equations is one in which b(t) = 0. Theorem If A(t) is an n n matrix function that is continuous on the

  System, Linear, Differential, Equations, Linear systems of differential equations, Linear systems

Power series and Taylor series - University of Pennsylvania

1. Geometric and telescoping series The geometric series is X1 n=0 a nr n = a + ar + ar2 + ar3 + = a 1 r provided jrj<1 (when jrj 1 the series diverges). We often use partial fractions to detect telescoping series, for which we can calculate explicitly the partial sums S n. D. DeTurck Math 104 002 2018A: Series 3/42

  Series, Power, Geometric, Power series, Geometric series

Generalized Eigenvectors - University of Pennsylvania

Eigenvectors Math 240 De nition Computation and Properties Chains Facts about generalized eigenvectors The aim of generalized eigenvectors was to enlarge a set of linearly independent eigenvectors to make a basis. Are there always enough generalized eigenvectors to do so? Fact If is an eigenvalue of Awith algebraic multiplicity k, then nullity ...

  Eigenvectors

4 Number Theory I: Prime Numbers - Penn Math

Number theory is filled with questions of patterns and structure in whole numbers. One of the most important subsets of the natural numbers are the prime numbers, to which we now turn our attention. ... make up the matter of our universe. Unlike the periodic table of the elements, however, the list of prime numbers goes on indefinitely.

  Universe, Patterns

Linear Algebra Problems - Penn Math

g) The linear transformation T A: Rn!Rn de ned by Ais onto. h) The rank of Ais n. i) The adjoint, A, is invertible. j) detA6= 0. 14. Call a subset S of a vector space V a spanning set if Span(S) = V. Suppose that T: V !W is a linear map of vector spaces. a) Prove that a linear map T is 1-1 if and only if T sends linearly independent sets

  Linear, Transformation, Linear transformations

Combinatorics and counting - Penn Math

combinatorics and counting 3 Overview of formulas Every row in the table illustrates a type of counting problem, where the solution is given by the formula. Conversely, every problem is a combinatorial interpretation of the formula. In this context, a group of things means an unordered set. Problem Type Formula Choose a group of kobjects from ...

  Combinatorics

5.3 Planar Graphs and Euler’s Formula

5.3 Planar Graphs and Euler’s Formula Among the most ubiquitous graphs that arise in applications are those that can be drawn in the plane without edges crossing. For example, let’s revisit the example considered in Section 5.1 of the New York City subway system. We considered a graph in which vertices represent subway stops and edges represent

  Graph

Eigenvalues, Eigenvectors, and Diagonalization

Find all of the eigenvalues and eigenvectors of A= 1 1 0 1 : The characteristic polynomial is ( 1)2, so we have a single eigenvalue = 1 with algebraic multiplicity 2. The matrix A I= 0 1 0 0 has a one-dimensional null space spanned by the vector (1;0). Thus, the geometric multiplicity of this eigenvalue is 1.

  Eigenvalue, Eigenvalues and eigenvectors, Eigenvectors, 1 0 0 0, 1 1 1 0

11 Partial derivatives and multivariable chain rule

the chain rule gives df dx = @f @x + @f @y ·y0. (11.3) The notation really makes a di↵erence here. Both df /dx and @f/@x appear in the equation and they are not the same thing! Derivative along an explicitly parametrized curve One common application of the multivariate chain rule is when a point varies along

  Chain, Partial, Derivatives, Partial derivatives

Algorithms and Complexity

is the study of computational complexity. Naturally, we would expect that a computing problem for which millions of bits of input data are required would probably take longer than another problem that needs only a few items of input. So the time complexity of a calculation is measured by expressing the running time of the calculation as a ...

  Computational, Complexity, Computational complexity

IXL - IXL - IXL | Math, Language Arts, Science, Social ...

2 Set up your roster It’s quick and easy to get your students ready to explore all the fun on IXL. You can effortlessly track class and student progress with our interactive, real-time Analytics. Your students have access to unlimited math, language arts, science, social studies, and Spanish practice in thousands of topics! QUICK-START GUIDE

  Quick, Math, Easy, Quick and easy

CHAPTER 9 The Pythagorean Theorem - High School Math

Step 2 To locate the center of the square on the longer leg, draw its diagonals. Label the ... with compass and straightedge.It is also easy to create several examples using geometry software. Step 2As needed, remind students that the legs are ... A quick check demonstrates that it doesn’t hold for other triangles.

  Quick, Math, Easy, Theorem, Pythagorean, Pythagorean theorem

Matrix Algebra and Applications - UTEP

An easy way to remember this is to think of the acronym “RC” for “Row then Column.” quick Example Referring to the Entries of a Matrix There is a systematic way of referring to particular entries in a matrix. If i and j are num-bers, then the entry in the ith row and jth column of the matrix A is called the ijth entry of A.

  Quick, Easy

Using Writing in Mathematics to Deepen Student Learning

and history classes, but not in mathematics. Math classes previously relied on skill-building and conceptual understanding activities. Today, teachers are realizing that writing during a math lesson is more than just a way to document information; it is a way to deepen student learning and a tool for helping students gain new perspectives.

  Math

Math Olympiad Hardness Scale (MOHS) - Evan Chen

2 Speci cation Here is what each of the possible ratings means.1 Rating 0M: Sub-IMO. Problems rated 0 are too easy to use at IMO. I can often imagine such a problem could be solved by a strong student in an honors math class, even without olympiad training. Rating 5M: Very easy. This is the easiest rating which could actually appear while

  Math, Easy

UDL Guidelines Checklist

Checkpoint 2.2 - Clarify syntax and structure Clarify unfamiliar syntax (in language or in math formulas) or underlying structure (in diagrams, graphs, illustrations, extended expositions or narratives) through alternatives that: o Highlight structural relations or make them more explicit o Make connections to previously learned structures

  Math

Determinants, part III Math 130 Linear Algebra

2x + 3y 4z = 2 3x 2y + 5z = 7 First, compute the determinant of the 3 3 co-e cient matrix. = 1 1 3 2 3 4 3 2 5 = 54 Next, replace the rst column by the constant vec-tor, and compute that determinant. x = 6 1 3 2 3 4 7 2 5 = 27 Then in the unique solution, x = x= = 1 2. Next, replace the second column by the constant vector, and compute that ...

  Determinants, Math

The RSA Algorithm

E(D(M)) = M: (2) c) Eand Dare easy to compute. d) The publicity of Edoes not compromise the secrecy of D, meaning you cannot easily gure out Dfrom E. With a given E, we are still not given an e cient way of computing D. If C= E(M) is the ciphertext, then trying to gure out Dby trying to satisfy an Min E(M) = Cis unreasonably di cult: the number

  Easy

Solutions to Problems on the Newton-Raphson Method

2. Let f(x)=x2 −a. Show that the Newton Method leads to the recur-rence x n+1 = 1 2 x n+ a x n : Heron of Alexandria (60 CE?) used a pre-algebra version of the above recurrence. It is still at the heart of computer algorithms for nding square roots. Solution:Wehavef(x)=2x. The Newton Method therefore leads to the recurrence x n+1 = x n− f(x ...