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Review sheet for exam 2  Math 291, Fall, 2007
Definitions
: Subspace, linearly dependent/independent sets of vectors, basis, span, null space,
row space, column space, dimension, rank, nullity, linear transformations, Ker(
f
), Range(
f
), eigen
values and eigenvectors, characteristic polynomial of the matrix
A
, diagonalizable matrix, coordi
nates of the vector
v
in some basis, change of basis matrix from
{
e
1
, . . . ,
e
n
}
to
{
f
1
, . . . ,
f
n
}
.
Main results
:
•
Let
A
be
m
×
n
and
f
A
:
R
n
→
R
m
the corresponding linear transformation. Then Ker(
f
A
)
is a subspace of
R
n
and Range(
f
A
) is a subspace of
R
m
. Moreover,
x
∈
Ker(
f
A
)
⇐⇒
A
x
=
0
⇐⇒
x
is in the null space of
A
⇐⇒
x
is a solution to the homogeneous equation.
y
∈
Range(
f
A
)
⇐⇒
the linear system of equations
A
x
=
y
has a solution
x
⇐⇒
y
is
a linear combination of the columns of
A
(namely
A
x
)
⇐⇒
y
∈
the column space of
A
.
That is, Range(
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This note was uploaded on 05/23/2008 for the course MATH 290/291 taught by Professor Lerner during the Spring '08 term at Kansas.
 Spring '08
 LERNER
 Linear Algebra, Algebra, Transformations, Vectors, Sets

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