7—Operators and Matrices 187Compute the components as usual.
M~e 1 =(
1 2
3 4
)(
1
0
)
=
(
1
3
)
= 1~e 1 + 3~e 2This says that the first column of the components ofMin this basis are( 1 3 ). What else would you expect?
Now select a different basis.
~e 1 =(
1
1
)
, ~e 2 =(
1
− 1
)
Again compute the component.
M~e 1 =(
1 2
3 4
)(
1
1
)
=
(
3
7
)
= 5
(
1
1
)
− 2
(
1
− 1
)
= 5~e 1 − 2 ~e 2M~e 2 =(
1 2
3 4
)(
1
− 1
)
=
(
− 1
− 1
)
=− 1 ~e 1The components ofMin this basis are (
5 − 1
−2 0
)
It doesn’t look at all the same, but it represents the same operator. Does this matrix have the same determinant,
using Eq. ( 26 )?
Determinant of Composition
If you do one linear transformation followed by another one, that is the composition of the two functions, each
operator will then have its own determinant. What is the determinant of the composition? Let the operators be
FandG. One of them changes areas by a scale factordet(F)and the other ratio of areas isdet(G). If you use
the composition of the two functions,FGorGF, the overall ratio of areas from the start to the finish will be the
same:det(FG) = det(GF). Recall that the the determinant measures the ratio of areas for any input area, not
just a square; it can be a parallelogram. The overall ratio of the product of the individual ratios,det(F) det(G).
The product of these two numbers is the total ratio of a new area to the original area and it is independent of
the order ofFandG, so the determinant of the composition of the functions is also independent of order.
Now what about the statement that the definition of the determinant doesn’t depend on the original area
that you start with. To show this takes a couple of steps. First, start with a square that’s not at the origin. You