Unit I: Systems of Linear Equations
1. Homogeneous Systems
vectors, the vector space
R
n
, addition of vectors, multiplication of vectors by scalars,
linear combinations of vectors
homogeneous system of linear equations, coeﬃcient matrix of the system
GaussJordan elimination algorithm, pivot, basic (leading) variable, free variable, form
of general solution
Theorem 1.
If a homogeneous system has more unknowns than equations, then there is
a nontrivial (nonzero) solution of the system.
Gaussian elimination with backsolving
The GaussJordan elimination algorithm provides a clear idea of the structure of the so
lution set for the system. However for actually solving the system, it is usually more eﬃcient
to use Gaussian elimination with backsolving, as it requires fewer arithmetic operations.
2. Matrix Notation
Let
A
be an
m
×
n
matrix. We write
A
= [
a
1
,...,
a
n
], where
a
1
,...,
a
n
are the column
vectors of
A
, and we deﬁne
A
x
=
a
1
x
1
+
a
2
x
2
+
···
+
a
n
x
n
. Thus
A
x
is a linear combination
of the columns of
A
.
With this notation, the matrix form of the linear homogeneous system of equations
corresponding to
A
is simply
A
x
=
0
.
The matrix
A
operates linearly on vectors:
A
(
x
+
y
) =
A
x
+
A
y
and
A
(
c
x
) =
cA
(
x
).
Linearity is equivalent to:
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 Spring '08
 lee
 Linear Algebra, Linear Equations, Addition, Equations, Multiplication, Vectors, Scalar, Systems Of Linear Equations, Vector Space, elementary row operations, rowreduced echelon form

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