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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
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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 an
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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.
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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
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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
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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 involv
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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 num
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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 matri
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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
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5.7 General Inner Products
1.
(a)
True
False
False
False
P1
P2
P3
P4
(b)
True
True
True
False
3. Use the fact that z
product.
Al
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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 augmente
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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 =
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1.3 Homogeneous Systems
1. Notice that only the coecient matrix is given. Transform the augmented matrices to reduced row-echelo
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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. P
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1.2 Linear Equations
1. (a) Simply verify that X satises the given equations.
(1) 2(2) + 3(0) + (0) = 3, and 2(1) (2) + 3(0) (0)
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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, s
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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. Eac