Is the mother sinful if she prevents her committed daughter from traveling to seek knowledge far from home, and what is the advice directed to the daughter?

It is not permissible for a woman to travel, whether for seeking knowledge or for any other purpose, without a mahram (definition), regardless of whether the journey is long or short. This is based on the Prophet's (peace and blessings be upon him) saying: "A woman should not travel except with a mahram."

If the lessons are within the city, it is permissible for her to attend them. The mother has the right to prevent her from spending the night outside the house or from anything that she fears might lead to fitna (strife/temptation) or evil. However, if the place and the companions are safe, and the matter does not require an overnight stay or travel, there is no reason to prevent her, as seeking knowledge is a great good.

If the place is safe and the company is righteous, her brother can travel with her to drop her off and bring her back, while exercising caution and offering help to anyone who wishes to seek knowledge and obey Allah.