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, x2, x3) = (3x2-x3, x1-4x2, x3) is a linear transformation and find itsstandard matrix.3. (25 pts.)Find the eigenvalues and eigenspaces of11202-100-1.4. (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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