I threw together a rough idea for a Matrix/Linda Danvers Supergirl show that doesn't require any extensive knowledge or use of Krypton, Superman, etc. that I've decided to share in brief:
Linda Lee Danvers wakes up severely injured and with no memory of the previous 48 hours of her life. As she starts to recover, she begins to experience brief 'vision flashes' and slowly starts to discover that she's been turned into a superhero. The character would 'morph' into the classic image of blonde Supergirl because of a brief image in one of these 'flashes', and the Matrix character would exist in protoplasmic form as the 'source' of the character's powers, as well as in the form of an 'avatar' that you'd occasionally see in 'flashes' as well and would provide the inspiration for the blonde, blue-eyed 'public image' that Linda shows to the world as a superheroine.
I don't want to go into any more detail than that because I have my own explicit plans for this idea, but I did want to offer it up as an example as to how you could do a Linda Danvers/Matrix Supergirl series without needing to mention Krypton, Superman, the Kents, Lex, etc.