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Unformatted text preview: Prof. Ming Gu, 861 Evans, tel: 23145
Oﬃce Hours: MWF 3:004:00PM
Email: [email protected]
http://www.math.berkeley.edu/∼mgu/MA54 Math54 Sample Midterm I, Fall 2007 This is a closed book, closed notes exam. You need to justify every one of your
answers unless you are asked not to do so. Completely correct answers given without
justiﬁcation will receive little credit. Look over the whole exam to ﬁnd problems that
you can do quickly. You need not simplify your answers unless you are speciﬁcally
asked to do so. Hand in this exam before you leave.
Problem Maximum Score
1
2 19 6 19 Total Your SID: 19 5 Your GSI: 19 4 Your Name: 19 3 1. (5 Points) 5 100 Your Score Math54 Sample Midterm I, Fall 2007 2 2. (19 Points)
(a) Solve linear systems of equations A x = b, where 123 A= 1 2 4 211 6 and b = 7 .
4 (b) Consider linear systems of equations A x = b, where 124 A = 1 2 k2 211 6 and b = 3k .
4 For what values of k does the system have a unique solution? inﬁnite number of
solutions? no solution? Math54 Sample Midterm I, Fall 2007 3 3. (19 Points) Let P be the set of all functions of the form c0 + c1 sin(x) cos(x) + c2 cos2 (x) +
c3 sin2 (x), where the c’s are arbitrary real constants. It is known that P is a linear space
under the usual function addition and scalar multiplication. Find the dimension and a basis
for P . Math54 Sample Midterm I, Fall 2007 4 4. (19 Points) Let u1 , · · · , um be vectors in span{v1 , · · · , vk }; and let v1 , · · · , vk be vectors in
span{w1 , · · · , wn }. Show that u1 , · · · , um are vectors in span{w1 , · · · , wn }. Math54 Sample Midterm I, Fall 2007
5. (19 Points) If the image of an n × n matrix A is Rn , show that A must be invertible. 5 Math54 Sample Midterm I, Fall 2007 6 6. (19 Points) Find examples of n × n matrices A and B such that A, B are not invertible but
A + B is. ...
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This test prep was uploaded on 04/03/2008 for the course MATH 54 taught by Professor Chorin during the Fall '08 term at Berkeley.
 Fall '08
 Chorin
 Math

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