When can a relative pronoun be omitted?

A relative pronoun can be implied (is not used in the sentence but is understood by the speaker and the listener) when the meaning of the sentence is not affected. Examples:

'This is the kind I like.' (This is the kind that I like.)

'This is the place I bought my car.' (This is the place where I bought my car.)

Examples of sentences where the relative pronoun should not be omitted:

'Mrs. Lincoln is new is my math teacher. ' (Mrs. Lincoln who is new is my math teacher.)

'The car hit mine was a taxi.' (The car that hit mine was a taxi.)

'The assignment is due on Friday is not finished.' (The assignment which is due on Friday is not finished.)

These sentences are not as easily understood without the relative pronouns.