Скачать или смотреть Linear Algebra Lecture 71 | Defining Addition of Cosets

In this lecture (Lecture 71), we define the addition of cosets in a vector space.

We start by taking one element from the coset u + W and one from v + W,
and show that their sum always lands inside the coset (u + v) + W.

We also show the converse: every element of (u + v) + W can be written as a sum of elements from u + W and v + W.

This motivates the natural definition:

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

Finally, we identify a very important issue:
cosets have many different representatives, so we must check whether this definition is well-defined.
That will be the topic of the next lecture.

Topics covered:
• Adding elements from two cosets
• Why the sum always lands in a particular coset
• The natural definition of coset addition
• Motivation for well-definedness
• Preparing for quotient space

Next Lecture (72):
We verify that this addition is well-defined and build further toward the quotient vector space.