Thread: question about projection

Dears,
I need the solution of the following exercise:

Let V be a real vector space and E an idempotent linear operator on V, i.e.,
a projection. Prove that (I + E) is invertible. Find (I + E)^-1. (where I is the identity operator)

Best Regards

hint:

suppose A is a matrix such that (I + E)A = I.

then A + EA = I.

applying E to both sides we have:

E(A + EA) = E

EA + E²A = E

but E² = E (since E is a projection) thus:

2EA = E, or EA = (1/2)E.

now substitute this back in the equation

A + EA = I, and solve for A.