1. Prove that any linear map can
written as an
matrix. (Hint: let
. Express
as a linear combination of the canonical basis vectors
. Substitute that into
, then use linearity to rewrite the right-hand-side of this equation as a linear combination of vectors
. Now take the inner product of both sides of this equation with
. That should give you
for some matrix
.)
2. Prove that .
3. If and
are
-vectors the matrix
is known as a
rank-one perturbation of the indentity. Show that if
is nonsingular, then its inverse
has the form
for some scalar
, and give an expression
for
. For what
and
is
singular? If it is singular, what is
?
(Trefethen exercise 2.6).