"The doctor is alone."
Translation:Az orvos egyedül van.
You need van here, because egyedül is an adverb, and adverbs can't be the statement of a sentence. In this case you need a verb, which is in this case the conjugated form of lenni.
Just think of this sentence like:
The doctor xxxx alone. Nice but what does the doctor do alone? That's why you need lenni.
Beyond this, adjectives, nouns, pronouns can be the statement, but an adverb can't take this role.
"Van" (or in the case of plural "vannak") is indeed needed here - as Woolfool correctly explained. "Egyedül" is an Adverb fulfilling the role of Adverbial (describing an aspect of circumstances of the subject) within clause type Subject - Verb - Adverbial, and in this clause type the third person singular and plural forms of "lenni" ("van" and "vannak") are required, as opposed to clause type Subject - Verb - Subject Complement, where the Verb "van" and "vannak" are omitted, both when the complement is a 1. noun phrase or 2. adjective phrase (eg 1. He is a doctor - ő orvos or 2. He is tall, ő magas.)
Please note, that the first and second person forms of "lenni" (vagyok, vagy, vagyunk, vagytok) are also required in the Subject- Verb - Subject Complement clause type - only the third person forms of "lenni" are omitted.
On this point: In fact "alone" - the English equivalent of "egyedül" - is not an Adjective, but an Adverb as well. In this context, the word "lonely" springs to mind as an Adjective. If "alone" was an Adjective, then it could also be used as premodifier of a Noun, such as "lonely" in "lonely doctor". But the phrase "alone doctor" would be unacceptable.
"Piros a lámpa" is just a description of the current state. Of course context matters a lot, if I am in the room, I will know that whoever said that is referring to a lamp in the room which is red, and if I am in a car then I will know that someone is referring to a traffic light that it is currently red.
To say that a traffic light turned red, that would rather be "A lámpa pirosra váltott".
Does this help ?
I have a problem with the general architecture of the presentation of the Hungarian language. How come there is only one speed of the soundtrack. Unfortunately it is only the fastest one which does give a chance to hear properly the phrases. It would be nice to also get a slower version of the said phrases. This leads me to make a lot of mistakes, hence to stagger in my learning and comprehension of the beautiful and mysterious Hungarian language.
Okay, I put "Az orvos van egyedül" for this answer because two answers ago, the discussion told me that "Soha nem vagyok egyedül" was the wrong order because the meaning was strange. Here you are telling me that it is the opposite order. Is there a way I can remember the correct order, please?
From what I understand, the difference comes from what the sentence is saying. In "Az orvos edgyedül van", we are emphasizing that the doctor is alone. In the sentence "Soha nem vagyok egyedül" we are emphasizing that I am never alone. If you were to say "Soha nem egyedül vagyok" it would sound like you are saying "I am never alone" while implying there is someone else always with you.