Question

The last one for tonight
If A, B, C, D are linear transformations (all on the same vector space)and if both (A+B) and (A-B) are invertible, then there exist linear transformations, X and Y such that
AX + BY = C
and
BX +AY = D
Please, I don't get what I am supposed to do

Answer 1

Do they ask me to prove there exists X and Y like that?

Answer 2

yess m but u have the key, start from
invertible,

Answer 3

hehehe. not too far .

Answer 4

so got it or should we do it here ?

Answer 5

Please, give me some steps. I am so tired for those stuff

Answer 6

start by adding & subtracting the 2 equations

Answer 7

@eashy We can't use those equations, they are conclusion we have to prove, right?

Answer 8

invertible
gives u
(A+B)(A-B)=I
G cong (G)=I
G=cong ^-1 (G)
let me remember what we conclude from that

Answer 9

we can assume the equations are true for now, and then use the equations to find explicit solutions for X and Y

Answer 10

thereby proving the equations

Answer 11

@ikram002p (A+B) and (A-B) invertible do not mean (A+B) is inverse of (A-B)

Answer 12

1) add the equations: (A+B)(X+Y)=C+D
2) subtract the equations: (A-B)(X-Y)=C-D
3) solve for X,Y

Answer 13

@eashy oh yes, I got what you mean, hehehe

Answer 14

hought it mean
(A+B) (A-B)=(A-B)(A+B)=I
right ??!

Answer 15

@ikram002p I don't think so, consider (A+B) = T, so \(T^{-1} \) totally different from inverse of (A-B)

Answer 16

hmm ok !
my bad lol

Answer 17

No problem, You help me a lot. hihihi

Answer 18

@eashy Thank you so much. It's 2:00 am, now I have to go to bed.

Answer 19

lol its 6:54 here

Answer 20

where are you from?