# Using this handout a second major step compare the

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using this handout (a second major step), compare the given problems below with something we have done before (or on HW): quite often you will see similarities. These problems do not encompass everything that could appear on the midterm. They are simply a collection of past exam problems. Problem 1. Calculate the following determinant. Include all calculations. Be clear and organized. det 7 1 1 1 1 7 1 1 1 1 7 1 1 1 1 7 . Problem 2. What is the smallest n for which the set { I 3 , A, A 2 , . . . , A n } is linearly dependent in the space of matrices M 3 × 3 ( R ) for any 3 × 3 real matrix? Explain. Problem 3. Let A be an n × n matrix with complex entries ( n 2) such that det( A - λI ) = (7 - λ ) n . (a) Is A necessarily diagonalizable? Prove or give a counterexample. (b) Prove that A is invertible. Problem 4. Let T and U be two linear opeartors on a finite-dimensional space V . Suppose that T and U commute with each other; i.e. TU = UT . (a) What is a U -invariant subspace of V ? Define it. (b) If E 5 ( T ) is the eigenspace of T corresponding to the eigenvalue λ = 5, prove that E 5 ( T ) is a U -invariant subspace of V . Problem 5 Consider the linear operator T : P 3 ( R ) P 3 ( R ) given by T ( f ( x )) = f ( x ) + f (1) · (1 + x 3 ) . (a) Find char T ( λ ). Factor and simplify the characteristic polynomial as much as possible. (b) Diagonalize T ; i.e. find a diagonal matrix D and a basis β for P 3 ( R ) such that [ T ] β = D. Problem 6. Consider the matrix A = 0 1 + i 2 - 2 i 0 1 - i 3 4 i 0 ! . (a) Calculate det A . Make sure you simplify your final answer. (b) If char A ( t ) = at 3 + bt 2 + ct + d , what are the coefficients a, b, and d equal to? Show all relevant calculations and explain. Problem 7. We would like to use Linear Algebra to find a direct formula for the Fibonacci sequence a n = a n - 1 + a n - 2 for n 2 with a 0 = 0 , and a 1 = 1. (a) Let ~v n = a n a n +1 for all n 0. Find a 2 × 2 matrix A such that ~v n +1 = A~v n for all n 0. Justify your answer! (b) For any n 0, express ~v n in terms of the matrix A and the vector ~v 0 . Justify your answer (c) How does the direct formula for a n depend on the eigenvalues λ 1 and λ 2 of A ? You do not have to diagonalize A or show the full direct formula for the Fibonacci sequences. It suffices to show where/how the eigenvalues appear in this direct formula. (d) To find the direct formula for the Fibonacci sequence, do we need to just find a diagonal matrix D similar to A , or do we also need to find an eigenbasis for A ? Explain briefly. 5
5 True/False Questions for Review Problem 8. Let T : R 3 lin. R 3 with matrix A = 0 1 6 - 1 0 5 0 0 4 ! in the standard basis E = { ~e 1 ,~e 2 ,~e 3 } . (a) Find a 2-dimensional T -invariant subspace W of R 3 . Give its basis and explain why it is T -invariant. (b) Is the subspace W you found in part (a) also T -cyclic? Explain why it is or why it isn’t. (c) Does W have a 1-dimensional T -invariant subspace U ? Why or why not? Problem 9. Let N be an n × n matrix that is nilpotent; i.e., N k = O for some integer k 1.
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