In the text below, should it be "Linda's starting" or "Linda starting"?
Linda changed her attitude in the middle of the second semester and started diligently memorizing new words and phrases. That, however, did not result in Linda's starting to make attempts to converse.
A usable grammatical pattern:
This follows the rule of the possessive pronoun plus a gerund
Etc.
Linda's is possessive, so: "Linda's starting to x" is right. This, is, therefore, a possessive.
[By the way, some people call these gerund-participles. Personally, I call them gerund nouns because they are also gerund adjectives.]
Of course, you don't have to use an apostrophe s but when you do, the above is why.
[1] That, however, did not result in [Linda starting to make attempts to converse].
[2] That, however, did not result in [Linda's starting to make attempts to converse].
These are both fine and have the same meaning except that [2] is more formal than [1]. In both examples, the bracketed gerund-participial clause functions as complement of the preposition "in".
The plain case "Linda" in [1] is just as correct as the genitive Linda's in [2] and functions as subject of their respective clauses.
There is no reason to consider the bracketed element in [2] a noun phrase; rather, it is better to treat the stylistic alternation between [1] and [2] as simply a matter of the case of the subject.
One reason for the clausal analysis is that if "Linda’s" really was a genitive determiner of the NP "starting ….", then it would not be freely replaceable with a plain case noun like "Linda", whereas it clearly is since [1] is unquestionably grammatical. The conclusion is that "starting" is a verb in both examples, and that "Linda" / "Linda's" is subject, with the verb phrase "starting to make attempts to converse" as predicate.
The short answer is yes, you do need the possessive noun with the gerund. If you didn't, then it wouldn't be a gerund! Your example works either way, with the gerund or without.
A gerund is a verb as a noun, so it is treated as a noun in every way. You wouldn't say "Linda car" to show that Linda possessed a car. In your example "starting" is being used as a noun because it is a fixed event and spoken of as if it belongs to her.
Just remember that the 'apostrophe s' doesn't make the verb that follows a gerund. It could be a contraction, for example "Linda's eating the cake" means "Linda is eating the cake".
For the sake of having a visible case change, let's replace "Linda" with "Bob":
Bob changed his attitude.
1) That did not result in his starting to make attempts.
2) That did not result in him starting to make attempts.
Under a traditional analysis, gerunds and participles form phrases, not clauses. They do not have subjects. This is supported by the observation that "he starting to make attempts" is ungrammatical.
Above, we have two grammatically sound options. In the first, the simple object of the preposition "in" is "starting". The gerund phrase "starting to make attempts" is modified by the attributive genitive pronoun "his". In the second, the simple object is "him". The participial phrase "starting to make attempts" modifies the objective pronoun.
The question is which simple object, if either, is more sensible. Which better represents this particular lack of result: the action associated with the man, or the man in that condition?
To my sensibilities and in the absence of further context, the lack of action is more relevant. This author made the same choice that I would be likely to make.
The simplest thing to do is to assume that the original is as it should be, that it expresses the author's intent, that the lack of action has the greater relevance in the author's mind. The other option is just as valid grammatically, but it places the emphasis on a different constituent.
