zyugyougo, sonomama tosyokan ni ikimasu.

授業後 is a set expression meaning after class.

• 授業 = class / lesson
• 後（ご / あと） = after

Here, 後 is attached directly to the noun to make a compound noun:

• 授業後 = after class
• 食後 = after a meal
• 放課後 = after school

So there is no particle between 授業 and 後 because they form one expression.

A similar alternative is 授業の後, which also means after class. That version is a little more transparently grammatical for beginners:

• 授業の後、そのまま図書館に行きます。

Both are natural.

In this sentence, そのまま means something like:

• just like that
• without doing anything else first
• straight
• right away

So the nuance is:

After class, I go straight to the library
or
After class, I go to the library without stopping anywhere else first.

It often suggests that no change or interruption happens in between. For example, the speaker does not go home first, does not change plans, and does not stop somewhere else.

Because そのまま is functioning adverbially here.

It modifies the verb 行きます and describes how the action is done:

• そのまま行きます = go as-is / go straight

Many adverb-like expressions in Japanese do not need a particle. So this is normal.

You may sometimes see そのままで, but that has a slightly different feel and is used in different structures. In this sentence, そのまま by itself is the natural choice.

に marks the destination of motion verbs like:

• 行く = to go
• 来る = to come
• 帰る = to return

So:

• 図書館に行きます = go to the library

This is one of the most basic uses of に.

Yes. You could say:

• 図書館へ行きます

Both mean go to the library.

The rough difference is:

• に emphasizes the destination as a point
• へ emphasizes the direction toward that destination

In many everyday sentences, the difference is small, and either one is fine. In this sentence, に is very natural and common.

Because Japanese sentences normally place the main verb at the end.

English often follows subject + verb + object/complements, but Japanese typically puts everything before the verb, and the verb comes last.

So the structure here is roughly:

• 授業後 = after class
• そのまま = straight / just like that
• 図書館に = to the library
• 行きます = go

That final verb is what completes the sentence.

Japanese often omits the subject when it is understood from context.

So this sentence could mean:

• I go straight to the library after class
• We go straight to the library after class
• (someone) goes straight to the library after class

In many textbook situations, the natural assumption is I.

This is very normal in Japanese. If the subject is obvious, it is usually left unsaid.

行きます is the polite non-past form.

In Japanese, the non-past can express:

• a future action: I will go
• a habitual action: I go

So this sentence could mean either:

• After class, I’ll go straight to the library or
• After class, I go straight to the library

The context tells you which one is intended.

行きます is polite speech, using the -ます form.

Compare:

• 行きます = polite
• 行く = plain / dictionary form

So this sentence is appropriate in normal polite conversation, classroom Japanese, or when speaking respectfully but not formally stiff.

The comma shows a pause and helps readability.

Here, 授業後 is a time expression at the beginning of the sentence, so a comma is often used after it:

• 授業後、そのまま図書館に行きます。

The comma is helpful, but in many cases it is not absolutely required. Japanese punctuation can be more flexible than English punctuation.

No. Normal Japanese is usually written without spaces between words.

The natural written sentence is:

授業後、そのまま図書館に行きます。

The spaces in your version are probably there just to help learners see the parts more clearly.

They are very close in meaning, and both are natural.

• 授業後 = compact, set-expression style
• 授業の後 = more analytically clear for learners

Examples:

• 授業後、そのまま図書館に行きます。
• 授業の後、そのまま図書館に行きます。

The first can feel slightly more concise or formal in style, but in everyday use both are common enough.

Yes, to some extent. Japanese word order is more flexible than English, as long as the particles make the role of each part clear.

For example, these are possible:

• 授業後、そのまま図書館に行きます。
• そのまま授業後に図書館に行きます。
• 図書館に、授業後そのまま行きます。

However, the original order is the most natural and easiest:

• time first: 授業後
• manner next: そのまま
• destination: 図書館に
• verb last: 行きます

So while reordering is possible, the original sentence sounds very standard.