Umm, in fact the lady is just a random woman who lodges him out of pure kindness. She has no relation whatsoever with Yvon.
She adresses Yvon with "vous", not "tu", making clear that they are not mother and son.
When they hear the footsteps she tells him it's HER sister and brother-in-law:
"C'est ma soeur et mon beau-frère, les parents du petit"
(It's my sister and my brother-in-law, the boy's parents).
And then she says:
"J'ai aussi une autre soeur, qui ne revient pas tous les soirs"
(I also have another sister, who doesn't sleep here every night).
This is barely an info that a mother would tell her son only after he's a grown-up, isn't it, that she has another sister;)?
In fact, the old man is HER father; when the lady brings Yvon his morning coffee and the old man stops her, she calls him "Papa". The old man is angry because she lodges a stranger, so he says, "tu ne le connais pas" ("you don't know him"), calls her mad and slaps her.
I suspect the old man is the lady's father-in-law, because later she tells Yvon:
"Ce n'est qu'après la morte de mon mari qu'il s'est mis à boire, et qu'il a perdu ses élèves"
(It's only after the death of my husband that he started drinking and lost his pupil).
I still don't understand tho' why he should kill the kind lady, whom he trusts enough to tell about his first murder and who nevertheless still agrees to lodge him!
"I can resist everything except temptations." - Oscar Wilde.
I took it to be just a random kind woman who took him in and it was a sign of just how fully corrupted Yvon is by the inhumanity of money and people that, despite the woman's kindness, he just emotionlessly murders the whole family for no particular reason. I was thinking that if this film was Hollywood-made, Yvon would have spent the second half of the movie getting revenge on all the people who screwed him over but instead Bresson has Yvon become even more evil than those betrayed him. The film shows random cruelties causing other random cruelties in a never-ending cycle.