Generator sets and the kernel formation (Ex)

Exercise 1: Let $\{x_1, x_2, \dots, x_k\}$$<wikitex>; '''Exercise 1:''' Let $\{x_1, x_2, \dots, x_k\}$ and $\{y_1, y_2, \dots, y_l\}$ be two sets of $\mathbb{Z}[\pi_1(X)]$-module generators for the kernel module $K_n(M)$. These are related by a sequence of elementaray operations. (See \cite{|Wall1999|Chapter 6}, and/or \cite{|Ranicki2002|Chapter 12}) # What is the effect on the kernel formation of $(f, b)$ of adjoining or deleting a zero? # What is the effect of permuting elements? # and of adding a new element which is a linear of combination of the others? </wikitex> <!-- == References == {{#RefList:}} --> [[Category:Exercises]] [[Category:Exercises without solution]]\{x_1, x_2, \dots, x_k\}$ and $\{y_1, y_2, \dots, y_l\}$$\{y_1, y_2, \dots, y_l\}$ be two sets of $\mathbb{Z}[\pi_1(X)]$$\mathbb{Z}[\pi_1(X)]$-module generators for the kernel module $K_n(M)$$K_n(M)$. These are related by a sequence of elementaray operations. (See \cite{|Wall1999|Chapter 6}, and/or \cite{|Ranicki2002|Chapter 12})

1. What is the effect on the kernel formation of $(f, b)$$(f, b)$ of adjoining or deleting a zero?
2. What is the effect of permuting elements?
3. and of adding a new element which is a linear of combination of the others?