Math 20F BenderFinal Exam3 PM June 15, 1995Please start each problem on a NEW PAGE.Remember to show your work.1. (25 pts.) Deﬁne the following:(a) an eigenvalue of then×nmatrixA,(b) the null space of anm×nmatrixA,(c) the dimension of a vector spaceV.2. (15 pts.)Show thatT(x1,x23)=(3x2-x31-4x23) is a linear transformation and ﬁnd itsstandard matrix.3. (25 pts.) Find the eigenvalues and eigenspaces of11 202-100-1.4. (20 pts.) LetWbe the span of the orthogonal vectors110-1,1011and0-11-1. Write3456as thesum of a vector in the subspaceWand a vector orthogonal toW.5. (20 pts.)Find the eigenvalues associated with the quadratic form 8x21+6x1x2and use them toclassify the form.6. (60 pts.) Suppose thatAis ann×nmatrix and thatRis the reduced row echelon form ofA.Youare givenRbutare not givenA. For each of the following explain why you can or cannot answer itgivenRbut notA.(a) DoesA-1exist?(b) What is a basis for NulA?(c) What is a basis for Col
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This note was uploaded on 06/18/2009 for the course MATH 20E taught by Professor Enright during the Winter '07 term at UCSD.