Question

show TB is linear if B^2=0
Tb(A)=(A+B)^2-(A+2B)(A-2B)

Answer 1

Multiply out the brackets, and you should come out wiser :D

Answer 2

TB(A)=A^2 +2BA+B^2-[A^2-4B^2] =2BA

Answer 3

when B^2=0

Answer 4

how does it show its linear though?

Answer 5

Anything non-linear would have higher powers of A or B.

Answer 6

Or indeed negative powers of -2 or lesser.

Answer 7

ok, thanks, so if B=\[\lceil 0 1 \rceil \lfloor 0 0 \rfloor\] (dont know how to do matrices) but with 0 1 on the top and 00 on the bottom) what is the rank of B?

Answer 8

\[\lceil 0 1 \rceil\]
\[\lfloor 0 0 \rfloor\]