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# Linear Algebra MATH 2270

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This 2 page Class Notes was uploaded by Miss Noel Mertz on Monday October 26, 2015. The Class Notes belongs to MATH 2270 at University of Utah taught by Nicholas Korevaar in Fall. Since its upload, it has received 24 views. For similar materials see /class/229924/math-2270-university-of-utah in Mathematics (M) at University of Utah.

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Date Created: 10/26/15

Math 22702 Second Exam Review Information October 26 2001 I have reserved JW B 208 tomorrow Saturday from 1 230 in the afternoon for a problem session The exam will cover section 34 4143 5155 In addition to being able to do the computations from these sections you should know key de nitions 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 rst exam there will also be some truefalse questions see e g those at the ends of chapters 3 and 5 One way to organize the topics is as follows Linear Spaces 34 43 also called vector spaces Definitions Linear space subspace Linear transformation domain codomain kernel image rank 111111it linear isomorphism linear combination span linear dependence independence basis dimension coordinates with respect to a basis matrix of a linear transformation Theorems results about dimension eg if dimVn 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 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 See how the matrix of a linear trans changes if you change basis Orthogonality Chapter 5 De nitions orthogonal magnitude unit vector orthonormal collection orthogonal complement to a subspace orthogonal projection angle correlation coefficient not on exam but interesting orthogonal transformation matrix transpose least squares solutions to Axb inner product spaces Theorems Pythagorean Theorem CauchySchwarz Inequality Any basis can be replaced with an orthnormal basis Gram Schmidt Algebra of the transpose operation symmetric antisymmetric algebra of orthogonal matrices QR factorization Computations coordinates when you have an orthonormal basis GramSchmidt orthogonal projections least squares solutions application to bestline fit for data matrix for orthogonal projection

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