MATH 3000 (Azoff) Fall
2009
Study
Guide for Chapters 1 and 2
Definitions
Rn
unit vector
line in Rn (parametric form)
hyperplane in Rn (cartesian or normal form)
linear combination of vectors
dot product of vectors
orthogonal (or perpendicular) vectors
angle between (non-zero) vectors
elementary row operations
echelon and reduced echelon matrices, pivot
rank of a matrix
singular, non-singular matrices
linear transformation
standard matrix of a linear transformation
elementary matrices
inverses and one-sided inverses of matrices
transpose
symmetric matrix
Basic Computations
operations on scalars, vectors and matrices
matrix reduction
solving systems of linear equations
translating problems involving lines, planes, curve fitting, linear
combinations and matrix equations to systems of linear equations
passing between cartesian and parametric representatios of lines and
planes
projections and angles between vectors
finding matrices of projections, reflections, and rotations in R2
computing inverses of matrices
Conceptual understanding
existence and uniquenes of solutions to Ax=b
good and bad features of vector and matrix operations, especially
multiplication
construction of examples and counterexamples
Study Guide and Worksheet for
Chapter 3
The basic concepts of this chapter are: linear combination, subspace,
span (noun and verb), linear dependence, basis, dimension, orthogonal
complement, row space, column space, null space, rank, vector space.
- Strive to understand the precise definitions of these terms and
their geometric connotations in R2 and R3;
compare with examples in text and class notes.
- Learn the computational techniques of the chapter, e.g., how to
use matrices to test for independence, etc, and be sure you understand
why they work.
- Learn how to use the theorems of the chapter to construct your
own examples, and to understand relationships among the basic concepts,
e.g., that every set of 4 vectors in R3 is dependent. (The
worksheet below should give you practice with this.)
- Review proof-type problems.
Fill in the following table. In each case, S refers to the set of all
linear combinations of the given vectors, and T refers to the
orthogonal complement of S.
Ambient
Space
|
Vectors
|
Independent?
|
Span ambient
space?
|
Basis for
ambient space?
|
Dim of S
|
Geom. Desc.
of S
|
Basis for T
|
R2
|
(1,2)
|
|
|
|
|
|
|
| R2 |
(1,2), (1,3)
|
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|
| R2 |
(1,2), (1,3), (2,5)
|
|
|
|
|
|
|
| R2 |
(1,2), (2,4), (3,6)
|
|
|
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|
| R2 |
(1,2), (2,4)
|
|
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| R3 |
(0,0,0)
|
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| R3 |
(1,1,1), (2,2,2)
|
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| R3 |
(1,1,1), (1,2,1), (1,3,1)
|
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| R3 |
(1,1,1), (1,1,2)
|
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| R3 |
(1,1,1), (1,2,1), (2,4,2)
|
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| R3 |
(1,1,1), (2,2,2), (4,4,4)
|
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|
Fill in
the blanks.
a) Let A be an m x n matrix of rank r. The columns of
A are independent if _____. The cols of A span Rm if
_____. The cols of A are a basis for Rm if _____.
b) A linearly independent set in R4 cannot have
more than _____ vectors.
c) Two distinct vectors in R4 which cannot
together be part of a basis for R4 are __________.
True or False ?
a) The vector (2,3,0,1) can be expressed as a linear combination
of (1,0,0,0) and (0,1,0,1).
b) The solution set of 2x+y-z=0 is a subspace of R3.
c) The solution set of 2x+y-z=5 is a subspace of R3.
d) If v1, v2, v3 are dependent
vectors, then there are c1,c2,c3 in R
so that c1v1+c2v2+c3v3=0
and |c1|+|c2|+|c3|>0.
e) Any three vectors in R7 are independent.
f) A set which spans R3 can contain 5 vectors.
g) The set S:={(x,y,z) | x+y=z } is a subspace of R3.
h) The set S:={(x,y,z) | x2+y2=z2
} is a subspace of R3.
i) The set S:={(x,y,z) | x2+y2+z2
= 0 } is a subspace of R3.
j) There is a set of 5 vectors in R7 which is
independent.
k) Any set of 4 vectors in R4 which is independent
forms a basis for R4.
l) There is a 3-dimensional subspace of R5.
Study
Guide for Section 3.6 and Chapters 4, 5, and 6
Section 3.6: Abstract Vector Spaces (up through Example 9, i.e., not responsible for inner products)
- Definition and subspaces
- Examples: spaces of matrices and spaces of functions
- Definitions of spanning, independence, and bases
- Testing independence in examples
- Finding bases in examples
Section 4.1: Inconsistent Systems and Projection onto a subspace V of Rn
Definition and underlying
concepts:
- ProjV b is the unique vector p in V
satisfying (b-p) perp V. Of
all vectors in
V, it is the one which is closest to b.
- review the
definition of subspace V of
Rn and its orthogonal
complement Vperp
- recall that
V+Vperp = Rn, and V
intersect Vperp
= {0}
- relation to
matrices: R(A) and N(A) are orthogonal complements of each other
Methods for computing ProjV b:
-
as in Chapter 1 when V is one-dimensional
-
via a matrix A whose column space is V, that is, ProjV b= Ax where x satisfies the normal equation ATAx=ATb
-
simplifying the above by using an orthogonal basis for V
- by formula when the columns of A are independent, that
is, ProjV
b= A(ATA)-1(ATb);
- by
computing projection onto Vperp first
Using projections to compute reflections
Least squares and Applications to data fitting
Section 4.2: Gram-Schmidt Process (only thru Example 4.2, that is, not including the discussion of the QR decomposition on pages 201-202)
- Definition of
orthogonal, orthonormal bases
- Constructing
them
- Their
advantages in computing coordinates, projections
Section 4.3:
Linear Transformations
-
Review of definition and standard matrix [T]std
-
Matrix of a linear transformation relative to a basis [T]beta
-
Change of basis formula
-
Using matrices relative to appropriate bases to compute formulas and standard matrices for projections and reflections
- Definition
and properties of similarity
-
Examples of linear transformations on spaces of functions
-
Kernels & images
Chapter 5 : Determinants
det (AB)=(det A)(det B)
det (AT)=det A
det A=0 if and only if A is singular
Section 6.1: Eigenvalues and Eigenvectors
-
Definition of eigenvalues, eigenvectors & eigenspaces
-
Characteristic polynomials and finding eigenvalues
-
Finding a basis for E(t) = null[A-tI]
Section 6.2: Diagonalizability
- Definition
and the equations
AP = PD ; A= PDP-1
- Using
eigenvalues and eigenvectors to construct D and P
- Eigenvectors
associated with distinct eigenvalues are independent
- Criteria for
diagonalizability
Section 6.3: Applications
- Computing powers of diagonalizable matrices
- Class attendance example and related Markov processes