Madame Gu thought about letting Gu Zhenkang marry the daughter of the Lin Family, which used to be an imperial business family.
The Lin Family and the Gu Family never had any connections. So the Gu Family must have made a huge plan so that Gu Zhenkang could get to know Ms. Lin and make her marry him without complaints.
The Gu Family was indeed a literati family, but Ms. Lin was not the best choice to be his wife. Nevertheless, Ms. Lin insisted on marrying this man, even though Lin Zeqing tried to stop it.
After Ms. Lin married into this family, the renovation of the Gu Mansion started. The Lin Family had paid for all the renovation, since the Gu Family were so poor at that time. That was how the Gu Mansion became what it was like now.
Therefore, the Gu Mansion should have been Ms. Lin's mansion, except for the name.