Math 2270 Exam 3
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University of Utah
Fall 2012
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Name:
e
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This is a 50 minute exan/Please show all your work, as a worked
problem is required for full points, and partial credit may be rewarded for
some work in the right direction.
1
1. (9 Points) The Four Subspaces
• (3 points) The nulispace is the orthogonal complement of the
If
• (3 points) an rn x n matrix has a d-dimensional column space,
then its left nullsp ace has dimension
n’i
—
• (3 points) For an m x ri matrix A calculate:
)) + dirri(N(A
T
)).
T
dim(C(A)) + dim(N(A)) + dim(C(A
2
2. (8 points) Orthogonal Complements
complement of the vector space
(/i\
spari( 2
-
Find a basis for the orthogonal
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3
3. (20 points) Projections - Find the projection of the vector b onto the
column space of the matrix A:
1 —1
11 1
011
1 0 1)
1
T
A
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4. (15 points) Linear Regression Find the least squares regression line
through the points (1, 0), (2, 1), and (3, 3).
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6. (8 points) Properties of Matrices Using the three fundamental prop
erties of the determinant (determinant of the identity is 1, switching
rows switches the sign, linearity for each row) prove that a matrix
with a row of Os has determinant 0.
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7. (15 points) Calculating Determinants
the matrix
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1 2 ( OZ I
1
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Calculate the determinant of
0 —1 1
10 2
1 —1 0
21 0
1
2
1
3
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8. (10 points) Other Ways of Calculating Determinants Calculate the de
terminant of the matrix
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(312
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using a cofactor expansion along row 3.
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