Version 087 EXAM02 gilbert (57245)
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when A has an LU -dec
khan (sak2454) HW02 gilbert (57245)
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Equivale
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MATRIX OPERATIONS, TRANSFORMATIONS
x
Given a vector
in
Rn
and an
m n matrix A we've learned to form a new vector, the product A x , using the
matrix-vector rule or the row-column rule for the product
VECTOR SPACES
Underlying so much of what's been done so far is the idea of linear combination - it's such a common, but
important, theme in mathematics and in all its applications that we convert it i
FACTORIZATION of MATRICES
Let's begin by looking at various decompositions of matrices, see also Matrix Factorization. In CS, these
decompositions are used to implement efficient matrix algorithms. In
INVERSE of a MATRIX
The term inverse you've met many times already with numbers: the number 1 is the inverse of
3
3
in the sense that
3
4
etc . We are now going to extend these ideas to matrices and
3
SETTING the SCENE: VECTORS and MATRICES
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Linear Algebra is a combination of concepts, computational techniques and applications - the interplay among
definitions, theore
SPANNING SETS, LINEAR INDEPENDENCE
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In earlier lectures we saw that lines and planes through the origin in
Line: cfw_ tv : < t < ,
given vectors
u, v
in
R3
could be ide
Linear Systems: REDUCED ROW ECHELON FORM
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The Gauss Elimination method in the previous lecture used elementary row operations such as
12
0 1
37
1
2
1
1
7
16
1 21
0 1
0
1
Linear Systems: GAUSS ELIMINATION
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Extending the equation
ax + by = c of a line in R2
and a plane
Definition 3.1: a LINEAR EQUATION in
Ax + By + Cz = D in R3
n variables
VECTORS in 2-space and 3-space
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Before starting on course material let's recall some ideas, notation and terminology you've met already. Then we can
start to reformulate
LINES and PLANES in 3-space
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y = mx + b is a familiar way of describing a line in 2-space as a linear equation. But this
equation doesn't allow for vertical lines like x
DIMENSION, BASES
We often speak of the plane as
2-space and the space around us as 3-space because we know that two
coordinates are needed to specify points in the plane, while three coordinates are n
GEOMETRIC TRANSFORMATIONS
So far we've used vectors and matrices for the most part to write a system of linear equations in more concise
form as a vector or matrix equation. This often enabled us to i
SINGULAR VALUE DECOMPOSITIONS: theory and examples
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The idea of a Diagonalization of a matrix has occurred already several times, in different forms, and for different
p
QUADRATIC FORMS
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Expressions of the form
3x2 + 2y2 ,
3x2 2y2 + 8xy ,
4 x2 + 3 x2 x2 2 x1 x2 + 8 x1 x3 4 x2 x3 ,
1
2
3
are called Quadratic
SYMMETRIC MATRICES, SPECTRAL THEOREM
Recall yet again:
The Transpose of an
m n matrix A is the n m
matrix
AT
obtained by interchanging
rows and columns: so
a12
a22
a1n
a2n
am1
A=
a11
a21
am2
amn
a21
a
Orthogonal Projections, GRAM-SCHMIDT
B = cfw_u1 , u2 , , up formed a basis for the
subspace W = Spancfw_ u1 , u2 , , up spanned by B, and that any vector x in W could be represented very simply
Theo
ORTHOGONALITY, ORTHONORMALITY
We've seen already how to re-interpret many properties of the plane and 3-space, as well as properties of vectors in the
plane and 3-space, in terms of matrices. What's m
khan (sak2454) HW14 gilbert (57245)
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1. TRUE
2. FALSE
10.
khan (sak2454) HW13 gilbert (57245)
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1
and
v2 =
x2
x2
=
1 4
.
khan (sak2454) HW06 gilbert (57245)
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5
2
, B=
1
1
5
A=
3
khan (sak2454) HW03 gilbert (57245)
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Every ma
khan (sak2454) HW06 gilbert (57245)
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Solve fo
khan (sak2454) HW05 gilbert (57245)
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002
4
3