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Unformatted text preview: Prelim 1 1. Math 294
Spring 2004
no calculators
answer + reason = credit
condensed printout—there were 5 pages a) (17 points) Find the number a for which the system
x1 + 3x3 − x4 = 20
x1 + 2x2 + 2x4 = a
x1 + 4x2 − 3x3 + 5x4 = 60 has solutions, and ﬁnd all solution vectors x.
b) (3 points) Give a deﬁnition: “ u1 , u2 , . . . , un are linearly independent iﬀ —”. 1 0 0 0
2 1 0 0
2. (20 points) Find the inverse C −1 if C = .
0 3 1 0
0 0 4 1
3. a) (8 points) Find a basis for the kernel of D, if D is the 5 by 5 matrix having a 1 in every position. −1
−1
1
0
0 −1 1 0
1
0
b) (8 points) In this part, let v1 = , v2 = , e1 = , e2 = , e3 = , and
1
1
0
0
1
1
−1
0
0
0 0
0
e4 = . Find a basis for R4 consisting of v1 and v2 together with some of the ek .
0
1
c) (4 points) Give at least 4 examples of vectors in the image, imA, where A is the matrix of coeﬃcients
in problem 1.
4. Let T : R2 → R2 be rotation clockwise by 45 degrees.
a) (10 points) Make a sketch to show why T has the linearity properties T (x + y) = T x + T y and
T (cx) = cT x.
1
b) (10 points) Find the matrix for T and evaluate T (
).
3
5. a) (10 points) Give an example of a 3 by 2 matrix A and a 2 by 3 matrix B for which BA = I, the 2
by 2 identity matrix.
b) (5 points) For your matrices of part a), verify that AB = I, the 3 by 3 identity matrix.
c) (5 points) Give a reason why AB could not be I no matter what you had picked in part a). 1 ...
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This note was uploaded on 04/04/2008 for the course MATH 2940 taught by Professor Hui during the Spring '05 term at Cornell University (Engineering School).
 Spring '05
 HUI
 Math, Linear Algebra, Algebra

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