linear-guest

1.4 So, What is a Matrix? 29
Example 10 of how a different matrix comes into the same linear algebra problem.
Another possible notational convention is to
denote a + bx + cx 2
as
⎛ ⎞
a
⎝b⎠
c
B ′ .
With this alternative notation
( ) ⎛ ⎞
a d
dx + 2 ⎝b
c
⎠B ′ =
( d
dx + 2 )
(a + bx + cx 2 )
= (b + 2cx) + (2a + 2bx + 2cx 2 ) = (2a + b) + (2b + 2c)x + 2cx 2
⎛ ⎞ ⎡⎛
⎞ ⎛ ⎞⎤
2a + b
2 1 0 a
= ⎝ 2b + 2c = ⎣⎝0 2 2⎠
⎝b⎠
2c
0 0 2 c
⎠B ′
⎦B ′ .
Notice that we have obtained a different matrix for the same linear function. The
equation we started with
⎡⎛
⎞ ⎛ ⎞⎤
( )
2 1 0 a
d
dx + 2 f = x + 1 ⇔ ⎣⎝0 2 2⎠
⎝b⎠
0 0 2 c
⎛
1
4
⎞
⇔
2a + b = 1
2b + 2c = 1
2c = 0
⎦B ′ =
⎛ ⎞
1
⎝1⎠
0
⎜ 1⎟
has the solution ⎝ 2⎠. Notice that we have obtained a different 3-vector for the
0
same vector, since in the notational convention B ′ this 3-vector represents 1 4 + 1 2 x.
B ′
One linear function can be represented (denoted) by a huge variety of
matrices. The representation only depends on how vectors are denoted as
n-vectors.
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