Math 103, Summer 2006
Midterm 2
August 3, 2006
MIDTERM 2
• Complete the following problems. You may use any result from class you like, but if you cite a theorem
be sure to verify the hypotheses are satisfied.
• This is a closed-book, closed-notes exam. No calculators or other electronic aids will be permitted.
• In order to receive full credit, please show all of your work and justify your answers. You do not need
to simplify your answers unless specifically instructed to do so.
• If you need extra room, use the back sides of each page. If you must use extra paper, make sure to
write your name on it and attach it to this exam. Do not unstaple or detach pages from this exam.
• Please sign the following:
“On my honor, I have neither given nor received any aid on this examination. I have furthermore abided by all other aspects of the honor code with
respect to this examination.”
Name:
Signature:
The following boxes are strictly for grading purposes. Please do not mark.
aschultz@stanford.edu
1
15 pts
2
15 pts
3
15 pts
4
20 pts
5
20 pts
6
15 pts
7
1 pt
Total
100 pts
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Page 1 of 9
Math 103, Summer 2006
Midterm 2
August 3, 2006
(1) (15 points) State the desired definition or complete the given sentence. Your answers should be clear
and consise.
(a) Define ‘eigenvector.’
(b) Define ‘orthogonal complement.’
(c) For an n × m matrix A, give a geometric interpretation of the quantity
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det(AT A).
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Math 103, Summer 2006
Midterm 2
August 3, 2006
(2) (15 points) Determine whether each statement is true or false. If the statement is true, cite your
reasoning. If it is false, provide a counterexample.
(a) A matrix whose columns are mutually orthogonal is an orthogonal matrix.
(b) For a 3 × 3 matrix A, if rank(A) = 2, then dim(im(A)⊥ ) = 1.
(c) If det(A) = 0 then 0 is an eigenvalue of A.
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Page 3 of 9
Math 103, Summer 2006
Midterm 2
August 3, 2006
(3) (15 points)
(a) Give an example of a matrix which preserves angles between vectors but does not preserve length.
(b) Give two matrices A and B which have the same (real) eigenvalues (counted with multiplicity) but
for which there exists an eigenvalue λ so that the geometric multiplicity of λ in A is not equal to
the geometric multiplicity of λ in B.
(c) Give an orthonormal basis of R2 which is not the standard basis.
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Math 103, Summer 2006
Midterm 2
(4) (20 points) Consider the matrix

1
 2
A=
 2
0
August 3, 2006

5
4 
.
7 
4
(a) Find an orthonormal basis for im(A).
(b) Give the QR-factorization of A.
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Page 5 of 9
Math 103, Summer 2006
Midterm 2
August 3, 2006
(c) What is the matrix of the linear operator projim(A) ? You do not need to simplify your expression.
(d) Explain geometrically why projim(A) has 0 and 1 as eigenvalues.
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Page 6 of 9
Math 103, Summer 2006
Midterm 2
(5) (20 points) Consider the matrix

1 3
B= 0 2
0 1
August 3, 2006

7
4 .
2
(a) Find the eigenvalues of B.
(b) For one of the eigenvalues λ above, find a basis for Eλ .
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Page 7 of 9
Math 103, Summer 2006
Midterm 2
August 3, 2006
(6) (15 points) For a square matrix A, prove that char(A) = char(AT ). This means that the eigenvalues
of a matrix A (counted with algebraic multiplicity) are the same as the eigenvalues of the matrix AT
(counted with algebraic multiplicity).
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Page 8 of 9
Math 103, Summer 2006
Midterm 2
August 3, 2006
(7) (1 bonus point) Give an example of a matrix A so that im(A) = ker(A).
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