A linking verb joins two words that refer to the same thing. For example, in the sentence 'John became a doctor', the words John and doctor refer to the same person.
In Russian, such verbs generally take the Instrumental case, e.g.
Ивђн стђл врачћм. Ivan became a doctor.
Verbs meaning 'be, become, be considered', etc. are considered to be linking verbs, although the concept is rather vague, and even the verb рабћтать might be considered to be a linking verb when it is synonymous with бІть, as illustrated in the first two examples below. In the final analysis, it is the dictionary that tells you what case to use with a particular verb.
Ћн рабћтал врачћм. He was a doctor (Lit. He worked as a doctor.)
Ћн бІл врачћм. He was a doctor.
Ћн стђл врачћм. He became a doctor.
Ћн считђется џмным человљком. He is considered (to be) an intelligent man.
Сђша хћчет бІть председђтелем. Sasha wants to be chairman.
There are various exceptions to this general rule, the most important of which is that to express 'am/is/are' (the present tense of 'be') you use no verb and no Instrumental case form, at least not in conversational Russian:
Ћн врђч. vs. Ћн рабћтает врачћм. He is a doctor vs. He works as a doctor.
In written Russian, on the other hand, there are various ways of expressing the present tense of 'be', the most important of which is the verb явлїться, which, like most linking verbs, governs the Instrumental case. The Instrumental case form often comes at the beginning of such sentences:
Сђмым глубћким ћзером в мќре явлїется Байкђл. Baikal is the deepest lake in the world.
Ћчень серьёзными явлїются вопрћсы, свїзанные с авђрией ђтомного ређктора. Problems having to do with an atomic reactor accident are very serious.
Some verbs, not in themselves linking verbs, take two objects which have a linking relationship. For example, in the sentence 'They elected Sasha [to be] chairman' the words Sasha and chairman refer to the same person. Note that the English linking verb 'to be' can be inserted between the two objects. In the corresponding Russian sentences the objects are Accusative and Instrumental, respectively.
In the following examples председђтелем is the Instrumental of председђтель and главћй is the Instrumental of главђ 'head'.
Сђшу вІбрали председђтелем. They elected Sasha chairman.
Президљнт назнђчил Джћнса главћй америкђнской делегђции. The president appointed Jones head of the American delegation.