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Lecture 4
4.1 Coordinatization of vectors
Let B = {
1
b
,
2
b
, …,
n
b
} be an ordered basis for a finite-dimensional vector space V, and
let
n
n
b
r
b
r
b
r
v
...
2
2
1
1
. Then [
r
1
,
r
2
, …,
r
n
] is the coordinate vector of
v
relative to
the ordered basis B, and is denoted by
B
v
.
Example: Find the coordinate vector of the polynomial p(x) = x
2
– 5x – 12 relative to the
ordered basis B = {x
2
, x, 1}.
The coordinate vector of
v
relative to an ordered basis of V is unique.
In general, each p(x) =
a
n
x
n
+
a
n – 1
x
n – 1
+ … +
a
1
x +
a
0
can be renamed by its coordinate
vector [
a
n
,
a
n – 1
, …,
a
1
,
a
0
] relative to the ordered basis B = {x
n
, x
n – 1
, …, x , 1}.
Coordinatization of a finite-dimensional vector space:
V: vector space, B =
n
b
b
b
...,
,
,
2
1
V
------------------------
R
n
v
B
v
w
v
B
B
B
w
v
w
v
r
v
(r
v
)
B
= r
B
v
Whenever the vectors in a vector space V can be renamed to make appear structurally
identical to a vector space W, we say V and W are isomorphic vector spaces
.
Every real vector space having a basis of n vectors is isomorphic to R
n
.
Example: Determine whether S = {1, x + x

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