Formula Sheet For Linear Algebra - Final Page 2
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1
2It follows that
1
proj
= A = A(A A)
A
,
so the standard matrix for the linear transformation proj
:
is
1
[proj ] = A(A A)
A .
6.5 Change of basis (for vectors)
= P [ ]
Solving for [ ] , we have
1
[ ] = P
.
To change directly from a nonstandard basis B to a nonstandard basis B , we have the
formula
1
[ ]
= P
P [ ] .
The same formulas hold in a general vector space V , as long as we set P = [
]
[
]
1
and replace
by [ ] for some basis E of V (usually chosen to be “standard”).
8.4 Matrices of general linear transformations
The matrix [T ]
can be computed as
[T ]
= [T (
)]
[T (
)]
[T (
)]
,
1
2
where B =
,
, . . . ,
.
1
2
[T ( )]
= [T ] [ ]
[T
T
]
= [T
]
[T
]
2
1
2
1
8.5 Similarity (change of basis for linear operators)
1
[T ] = P [T ] P
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