Midterm Exam I, Math 121 A
27 October, 2006
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Instructions: Put away any electronic devices, papers, notes, and books. Use only
a pencil and eraser (or a pen if you are daring). Write neatly and only on the paper
provided. Use the back o
Quiz VI, Math 121 A, 16 November, 2006
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1. (10 points): Let A be and m n matrix. Prove that if B can be obtained from A
by an elementary row operation, then B t can be obtained from At by an elementary
column operation.
Solution: If B c
Quiz V, Math 121 A, 9 November, 2006
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1. (10 points): Prove that if A and B are similar n n matrices, then tr(A) = tr(B ).
Solution: If A is similar to B , there exists an invertible n n matrix P such
that A = P 1BP . We have proven tha
Quiz IV, Math 121 A, 2 November, 2006
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1. (10 points): Find two matrices A and B such that AB = O but BA = O , where
O is the zero matrix.
Solution: The 2 2 matrices
A=
01
00
,
10
00
B=
will work. Note that AB = O , but
BA =
1
01
00
.
2
Quiz III, Math 121 A, 19 October, 2006
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1. (10 points): Give an example of a linear transformation T : R2 R2 such that
N(T) = R(T).
Solution: By the Dimension Theorem
rank(T) + nullity(T) = dim(V) = 2.
(1)
If N(T) = R(T) then they must
Quiz II, Math 121 A, 12 October, 2006
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1. (10 points): Prove that the set cfw_1, x, x2 , . . . xn is linearly independent in Pn (F ).
Solution: We consider an arbitrary linear combination of vectors from the set
= cfw_1, x, x2 , . . . xn and s
Quiz I, Math 121 A, 5 October, 2006
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1. (10 points): An m n matrix A is called upper triangular if all entries lying
below the diagonal entries are zero, that is, if Aij = 0 whenever i > j . Prove that
the upper triangular matrices form
Practice Final, Math 121 A
30 November, 2006
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Studying for the Final Exam: The nal exam will have eight to ten problems. The
problems will be drawn from (1) the seven quizzes, (2) the two previous practice exams,
and (3) the problems below. Note
Practice Exam II, Math 121 A
17 November, 2006
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Instructions: The actual midterm exam will have four to ve proof problems, that look
very much like a subset of the following eight. Complete these problems with the aid
of the book. The use of grou
Practice Exam I, Math 121 A
23 October, 2006
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ID:
Instructions: The actual midterm exam will have four to ve proof problems, that look
very much like a subset of the following eight. Complete these problems with the aid
of the book. The use of group
1
Math 121A Linear Algebra (Friday, Week 0)
5 to 7 quizzes (homework problems)
2 midterms
1 final exam
Chapters 1 to 4
Solutions, practice exams, and schedule on website.
Website: http:/math.uci.edu/~swise/classes/math_121a.html
Email: swise(at)mat
Final Exam, Math 121 A
8 December, 2006
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Solutions
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Instructions: Put away any electronic devices, papers, notes, and books. Use only
a pencil and eraser (or a pen if you are daring). Write neatly and only on the paper
provided. Use the back of th
Exam II, Math 121 A
22 November, 2006
Name:
Solutions
ID:
Instructions: Put away any electronic devices, papers, notes, and books. Use only
a pencil and eraser (or a pen if you are daring). Write neatly and only on the paper
provided. Use the back of the
Quiz VII, Math 121 A, 30 November, 2006
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1. (20 points): Determine if the following linear system has a solution.
x1 + 2x2 x3 = 1
2x1 + x2 + 2x3 = 3
x1 4x2 + 7x3 = 4
Solution: Consider the augmented system (A|b), where A is the coecient