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Linear Algebra Problems - Penn Math

Linear Algebra Problems Math 504 505 Jerry L. Kazdan Topics 1 Basics 14 Symplectic Maps 2 Linear Equations 15 Differential Equations 3 Linear Maps 16 Least Squares 4 Rank One Matrices 17 Markov Chains 5 Algebra of Matrices 18 The Exponential Map 6 Eigenvalues and Eigenvectors 19 Jordan Form 7 Inner Products and Quadratic Forms 20 Derivatives of Matrices 8 Norms and Metrics 21 Tridiagonal Matrices 9 Projections and Reflections 22 Block Matrices 10 Similar Matrices 23 Interpolation 11 Symmetric and Self-adjoint Maps 24 Dependence on Parameters 12 Orthogonal and Unitary Maps 25 Miscellaneous Problems 13 Normal Matrices The level of difficulty of these Problems varies wildly. Some are entirely appropriate for a high school course. Others definitely inappropriate. Although Problems are categorized by topics, this should not be taken very seriously. Many Problems fit equally well in several different topics. Note: To make this collection more stable no new Problems will be added in the future.

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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Generalized Eigenvectors - University of Pennsylvania

www2.math.upenn.edu

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Linear Systems of Differential Equations

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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.

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5.3 ORTHOGONAL TRANSFORMATIONS AND ORTHOGONAL …

staff.csie.ncu.edu.tw

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www.math.ucdavis.edu

Then 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 ...

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