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
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Linear Systems of Differential Equations
www2.math.upenn.eduLinear 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
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www2.math.upenn.edu5.3 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
Eigenvalues, Eigenvectors, and Diagonalization
www2.math.upenn.eduFind 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.
Power series and Taylor series - University of Pennsylvania
www2.math.upenn.edu1. 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
Generalized Eigenvectors - University of Pennsylvania
www2.math.upenn.eduEigenvectors 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 ...
11 Partial derivatives and multivariable chain rule
www2.math.upenn.eduthe 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
generatingfunctionology - Penn Math
www2.math.upenn.eduating functions can be simple and easy to handle even in cases where exact ... Generating functions can give stunningly quick deriva-tions of various probabilistic aspects of the problem that is repre-sented by your unknown sequence. (d) …
Algorithms and Complexity
www2.math.upenn.eduis 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 ...
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Combinatorics and counting - Penn Math
www2.math.upenn.educombinatorics 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 ...
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www.math.ucdavis.eduThen as a linear transformation, P i w iw T i = I n xes every vector, and thus must be the identity I n. De nition A matrix Pis orthogonal if P 1 = PT. Then to summarize, Theorem. A change of basis matrix P relating two orthonormal bases is an orthogonal matrix. i.e. P 1 = PT: Example Consider R3 with the orthonormal basis S= 8 >> < >>: u 1 = 0 ...
5.3 ORTHOGONAL TRANSFORMATIONS AND ORTHOGONAL …
staff.csie.ncu.edu.twA linear transformation T from Rn to Rn is called orthogonal if it preserves the length of vectors: kT(~x)k = k~xk, for all ~x in Rn. If T(~x) = A~x is an orthogonal transformation, we say that A is an orthogonal matrix. 1
Linear Transformation Exercises
web.ma.utexas.eduLinear Transformation Exercises Olena Bormashenko December 12, 2011 1. Determine whether the following functions are linear transformations. If they are, prove it; if not, provide a counterexample to one of the properties: (a) T : R2!R2, with T x y = x+ y y Solution: This IS a linear transformation. Let’s check the properties: