6. 8 points uppose t E „„,., IS a normal i•empotent operator t s. t ere exists E su that = 0). Prove that T = 0. 7. (8 points) Let T : V —¦ W. Prove that if T is onto then 7" is one-to-one.

Answer 6: if is a normal idempotent operator, and
Answer 6: if
nXn
TM
Î
is a normal idempotent operator, and
A = U*D where, U is a unitary operator.
This means that D can be composed so that D = U*A. in this case, a
simple matrix yields
0*00
000
UDUA
III
éùéùéù
êúêú=êú
êúêúêú
ëûëûëû
Also, in the same way, representation of T is valid.
For unique matrices, T = 0 * I *
0
0
A
I
éù
êú
êú
ëû
on every unique space
determined by T.
For T being a linear operator, on n-dimensional belonging to M such that
nXn
TM
Î
If and only if M is orthogonal to every image element of T.
Therefore, for every kernel value of T i.e.
0
k
T
=
, T = 0
Answer 7: If
:
TVW
®
in a linear transformation, assume that T is
one-to-one as T maps distinct in V to distinct vectors in W.
If
:
TVW
®
in a linear transformation, for all u and v in V,
T (u) = T(v) implies that u = v which makes
:
TVW
®
an one-on-
one function.
If...