operatorLonV
L:v∈V→Lv∈V
as multiplication by annbynmatrix:
v 1
v 2vn
→
L 11 L 12 ... L 1 n
L 21 L 22 ... L 2 nLn 1 Ln 2 ... Lnn
v 1
v 2vn
The reader should be warned that we will often not notationally distinguish
between a linear operatorLand its matrix with matrix entriesLjkwith respect
to some unspecified basis, since we are often interested in properties of operators
Lthat, for the corresponding matrix, are basis-independent (e.g., is the operator
or matrix invertible?). The invertible linear operators onVform a group under
composition, a group we will sometimes denoteGL(V), with “GL” indicating
“General Linear”. Choosing a basis identifies this group with the group of
invertible matrices, with group law matrix multiplication. ForV ndimensional,
we will denote this group byGL(n,R) in the real case,GL(n,C) in the complex
case.
Note that when working with vectors as linear combinations of basis vectors,
we can use matrix notation to write a linear transformation as
v→Lv=(
e 1 ··· en)
L 11 L 12 ... L 1 n
L 21 L 22 ... L 2 nLn 1 Ln 2 ... Lnn
v 1
v 2vn
We see from this that we can think of the transformed vector as we did above
in terms of transformed coefficientsvjwith respect to fixed basis vectors, but
also could leave thevjunchanged and transform the basis vectors. At times
we will want to use matrix notation to write formulas for how the basis vectors
transform in this way, and then will write
e 1
e 2en
→
L 11 L 21 ... Ln 1
L 12 L 22 ... Ln 2L 1 n L 2 n ... Lnn
e 1
e 2en
Note that putting the basis vectorsejin a column vector like this causes the
matrix forLto act on them by the transposed matrix.
4.2 Dual vector spaces
To any vector spaceVwe can associate a new vector space, its dual: