Скачать или смотреть Linear Algebra Lecture 72 | Well-Definedness of Coset Addition

In this lecture (Lecture 72), we prove that the addition of cosets is well-defined.

Earlier, we defined the operation:

(u + W) + (v + W) = (u + v) + W

But cosets can be represented in many different ways.
For example:
• u + W = u′ + W
• v + W = v′ + W

So we must check:

Does the result depend on the representatives we choose?

In this video, we prove that:

If
• u + W = u′ + W
• v + W = v′ + W

then
• (u + v) + W = (u′ + v′) + W.

This shows that coset addition is well-defined, and the operation does not depend on which representative we choose.

This is an essential step toward constructing the quotient space V / W.

In the next lecture, we will define scalar multiplication among cosets and then verify its well-definedness as well.