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22Apracticemidterm2

22Apracticemidterm2 - independence Check the class FAQ for...

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About the Midterm Here are some things you might want to study for the second midterm. The midterm will focus on the topics covered in class since the ﬁrst midterm (i.e., it won’t be cumulative), but of course, you will still be expected to know concepts from the ﬁrst part of the class that are applicable. LU decomposition (how to ﬁnd L and U , how to solve Ax = b using L and U , relation to elementary matrices. ..) Elementary matrices (know the three diﬀerent kinds, E i j ,R i ( λ ) ,S i j ( λ , what they are, what they do, how they relate to row reduction, how they relate to determinants and inverses. ..) Inverses (how to ﬁnd using row operations, Cramer’s Rule, what a cofactor is, what an adjoint is, how to use to solve systems of equations, how the determinant is involved, conditions for invertibility. ..) Determinants (how to ﬁnd a determinant (there are two ways), lots of theorems about determinants, permutations. ..) Subspaces, spanning sets, linear independence, basis (mostly know deﬁni- tions, be able to write down the span of a set of vectors, check for linear

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Unformatted text preview: independence. ..) Check the class FAQ for some more information on a few of these topics. Now, for a few sample problems: • Find the LU decomposition of M = 1 2 3 4 5 6 3 2 1 . Use the LU decompo-sition to solve the system MX = V , where V = 3 4 5 . • Suppose I use row operations to transform M = 1 2 3-1 2-3 3 2 1 into M = 9 6 3 0 4 0 1 2 3 . Find a matrix E such that EM = M . • Prove that if RREF ( M ) = 1 2 3 4 5 6 0 0 0 , then M cannot have an inverse. • What is the determinant of the inverse of the transpose of M = 1 0-3 2 0 0-1 8 3 1 0 2-1 0 ? 1 • Is 1 2 3 in the span of S = { 3 2 1 , -4 2 3 , -2 8 8 } ? Are the vectors in S linearly independent? Prove or disprove. 2...
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22Apracticemidterm2 - independence Check the class FAQ for...

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