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Chapter 5
VECTOR SPACES
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5.1 Examples and Basic Properties
1. (a) True. The zero vector.
(b) True. Every multiple of any nonzero vector.
(c) False. 1 is not in spancfw_1 x, 2 + x x2 .
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4.11 Singular Value Decomposition
1. Assume that A is n. Let 1 2 n 0 denote the singular values
m
of A where i = i for each i and the i are the eigenvalues of AT A. If we
write B = kA
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4.10 Complex Matrices
1. (a) False.
(b) False.
1
2i
1
2
2i
1
1
1
1
1
is Hermitian but not unitary.
is unitary but not Hermitian.
(c) True. If H = [hij ] is Hermitian, the condition H
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4.8 Quadratic Forms
1. (a) True. It is symmetric so the Principal Axis theorem applies.
11
(b) False.
.
02
2
1
(c) False.
.
1
2
(d) True. See the discussion following Theorem 3.
21
(e
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4.9 Linear Transformations
1. (a) False. Dene T : R2 R2 by T (X ) = AX where A =
1
0
0
0
, and take
0
.
1
(b) True. By Theorem 5.
(c) False. If T exists, then T (2e1 ) = 2T (e1 ), and
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4.6 Projections and Approximations
1. There are two methods for solving these types of problems.
Example 4, and the other involves systems of linear equations.
85
2
1
1
(a) X = 29 26
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4.7 Orthogonal Diagonalization
1. (a) True. Normalize any (nonzero) eigenvector.
1
1
2
(b) False. Consider the matrix
.
1
1
2
(c) True. The inverse of an orthogonal matrix is its tran
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4.3 Dimension
1. The dimension of a subspace is given by the number of vectors in a basis of the
subspace. So just count the number of vectors in the basis.
(a) 2.
(b) 3.
(c) 2.
(d) 3
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5.4 Isomorphisms and Matrices
1.
(a) False
(e) False
(b) True
(f) True
(c) False
(g) True
(d) False
(h) False
3. may help to look at the standard matrix of T. We have T (X ) = AX wher
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1.6 Elementary Matrices
1. You need to nd which row operation has been performed on the identity matrix
to yield the given matrix E. Let Ri denote i.
row
010
(a) The row operation is
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1.8 Markov Chains
1. Since there is only rain and shine, there is enough data given to ll in the
0.8
required Markov chain setup with initial condition S0 =
and transition
0.2
0.8 0.4
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1.5 Matrix Inverses
1. (A1 )1 is by denition a matrix B such that A1 B = I = BA1 , so B = A.
3. To nd A1 , you form the augmented matrix [A : I ], and you must transform A
to I ; othe
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1.7 LU-Factorization
1. To solve LU X = B , solve LY = B and then U X = Y .
[ILAW: Exploration 1.7.1.]
8
4
11
1
(a) Use B =
30 . Then LY = B gives Y = 2 , whence U X = Y
15
0
9
3
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1.3 Homogeneous Systems
1. Notice that only the coecient matrix is given. Transform the augmented matrices to reduced row-echelon form as usual.
[ILAW: Use Exploration 1.3.1], and rea
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References to Explorations and Lessons refer to the ILAW web tutorial.
Chapter 1
LINEAR EQUATIONS AND MATRICES
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1.1 Matrices
1. Put each table in matrix form, and use matrix operations
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5.6 Invariant Subspaces
1. (a) True. If v is in kerT then T [T (v)] = T (0) = 0 so T (v) is in kerT.
(b) True. If w is in imT, say w = T (v), then T (w) = T [T (v)] is also in imT.
(c
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4.2 Linear Independence
1. Follow the set up of Example 1 in the text and you solve the system of equations
that will arise. Each set of vectors is independent because the only soluti