U of U MATH 2270 - Final Exam Review Information

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Math 2270-1Final Exam Review InformationDecember 7, 2005 The usual problem session time and location, 9:40-10:30 Thursday December 8 (tomorrow) in LCB 121, will primarily be for homework due this week, but you can also bring up other questions then. I have will my usual office hours in LCB 204 next week (MW 1-1:50, T 11-11:50), as wellas an additional "special" time: Thursday, 9-11. Our exam is Friday morning, December 16th, 8-10 a.m., in our usual classroom JTB 120. I will let you stay until 10:30 a.m. Bring your winter wear since our classroom seems to be cold!!! The exam will be comprehensive, except for section 8.3 on singular value decomposition. Precisely, you can expect anything from chapters 1-8.2, as well as the Kolman material on conic sections/quadric surfaces, and the affine transformation material related to fractals. In addition to being able to do computations, you should know key definitions, the statements of the main theorems, and why they are true. The exam will be a mixture of computational and theoretical questions. As on the midterms, there will be some true-false questions drawn from course material. Exam material will be weighted towards topics which have not yet been tested, i.e. chapter 6-8 material. I am handing out the actual final exam I gave the last time I taught 2270. A copy, with solutions, will be posted on our web page. Of course, our exam will be different!Matrix algebra in Rn. (Chapters 1-2) (implicitly used in all other topics) Linear systems and matrix equations Ax=b intersecting hyperplane interpretation (linear system interpretation) linear combination of columns interpretation rref(A|b) and rref(A): how to compute, how to use. matrix transformations f(x)=Ax, and affine transformations f(x)=Ax+b. geometric properties (i.e. parallel lines get mapped to parallel lines, translations and scalings of any set are transformed into translations and scalings of the transformed set.) geometric transformations (scalings, rotations, projections, reflections, shears) inverse transformations and inverse matrices composition of transformations and matrix products matrix algebra (i.e. commutative, associative, distributive properties with addition and multiplication of matrices)Linear Spaces, (Chapters 3-4) :Chapter 3 was about Rn, and in Chapter 4 we generalized these ideas to general linear (vector) spaces. Definitions: Linear space subspace Linear transformation domain codomain kernel image rank nullity linear isomorphism linear combination span linear dependence, independence basis dimension coordinates with respect to a basis matrix of a linear transformation for a given basis Theorems: results about dimension: e.g. if dim(V)=n, then more than n vectors are ?, fewer than n vectors cannot ?, n linearly independent vectors automatically ?, n spanning vectors automatically are ? also, if a collection of vectors is dependent, it may be culled without decreasing the span; if a vector is not in the span of a collection of independent vectors, it may be added to the collection without destroying independence. the kernel and image of linear transformations are subspaces. rank plus nullity equals ? A linear transformation is an isomorphism if and only if ? Isomorphisms preserve ? Computations: Check if a set is a subspace (also, subspaces of Rn.) Check if a transformation is linear Find kernel, image, rank, nullity of a linear transformation Check if a set is a basis; check spanning and independence questions. Find a basis for a subspace Find coordinates with respect to a basis Find the matrix of a linear transformation, with respect to a basis Use the matrix of a linear transformation to understand kernel, image Compute how the matrix of a linear trans changes if you change basesOrthogonality (Chapter 5) Definitions: orthogonal magnitude unit vector orthonormal collection orthogonal complement to a subspace orthogonal projection angle correlation coefficient (not on exam, but interesting) orthogonal transformation, orthogonal matrix transpose least squares solutions to Ax=b inner product spaces Theorems Pythagorean Theorem Cauchy-Schwarz Inequality Any basis can be replaced with an orthnormal basis (Gram Schmidt) Algebra of the transpose operation symmetric, antisymmetric algebra of orthogonal matrices Orthogonal complement of the orthogonal complement of V is V! Computations find coordinates when you have an orthonormal basis (in any inner product space) Gram-Schmidt (in any inner product space) A=QR decomposition orthogonal projections (in any inner product space) least squares solutions application to best-line fit for data find bases for the four fundamental subspaces of a matrixDeterminants (Chapter 6) Definitions: recursive definition of determinant what is proof by induction? Theorems: determinant can be computed by expanding down any column or across any row (You don’t need to know the proof of this theorem!) effects of elementary row operations (or column ops) on the determinant of a matrix area/volume of parallelepipeds and determinants adjoint formula of the inverse, and Cramer’s rule determinant of product is product of determinants A is invertible if and only if its determinant is non-zero Computations: determinants by row ops or original definition inverse matrices via adjoint formula; Cramer’s rule for solving invertible systems. computing areas or volumes of parallelepipeds. the area or volume expansion factor of a linear transformationEigenvector concepts and applications (Chapters 7-8) Definitions: eigenvalue eigenvector characteristic polynomial eigenspace geometric and algebraic multiplicity of


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