Show that if AB is invertible so is B Let T be a linear tran

Show that if AB is invertible, so is B. Let T be a linear transformation that maps R^n onto R^n. Show that T^-1 exists and maps R^n onto R^n. Is T^-1 also one-to-one?

Solution

1.X is invertible if and only if detX0

If AB is invertible, detAB0.

But detAB=detAdetB0, so both detA and detB are nonzero, and so detB0

Therefore B are invertible.

2. T :Rⁿ --> Rⁿ is a linear transformation which is onto.

so for all Y in Rⁿ (co domain) there exist X in Rⁿ such that Y = T(X)

Now T(X) = 0    =>   X= 0 (because T is a LT)

THus T is one to one.

Thus T^-1 exist and ToT^-1 (X) =T^-1oT(X) = I(X).

Yes since T^-1 is also a LT ,so T^-1 (0)=0 .

so T^-1 is one to one.