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I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K).
I need help with Exercise 1.2.8 (a) (Chapter 1: Basics, page 3o) concerning $$K^n$$ as a $$K[T]$$-module ... ...
First, so that MHB readers will understand the relevant notation for the construction of $$K^n$$ as a $$K[T]$$-module, I am presenting the relevant text from B&K as follows:https://www.physicsforums.com/attachments/2997
Exercise 1.2.8 (a) (page 30) reads as follows:https://www.physicsforums.com/attachments/2998
https://www.physicsforums.com/attachments/2999So we are given that:
$$A$$ is an $$n \times n$$ matrix over a field $$K$$
$$U$$ is a subspace of $$K^n$$
$$M$$ is the $$K[T]$$-module obtained from the vector space of column vectors $$K^n$$
So we regard $$M=K^n$$ as a right module over $$K[T]$$
Addition in $$M=K^n$$ is normal column vector addition $$x+y$$ where $$x, y \in K^n$$ making $$M=K^n$$ an abelian group as required
Given an $$n \times n$$ matrix $$A$$ over $$K$$, a right action of $$f(T) \in K[T]$$ on $$M$$ is defined as follows:
$$xf(T) = xf_0 + Axf_1 + ... \ ... A^rxf_r
$$
where
$$f(T) = f_0 + f_1T + ... \ ... f_rT^r$$
We are required to show that:
------------------------------------------------
... a subspace $$U$$ of $$K^n$$ is a submodule $$L$$ of $$M$$ ...
if and only if
... $$AU \subseteq U$$
------------------------------------------------
Just reviewing the definitions of subspace and submodule in this context ...$$U$$ is a subspace of $$K^n$$ if $$U$$ is a subset of $$K^n$$ such that:
(1) $$0 \in U$$
(2) $$x, y \in U \Longrightarrow x + y \in U $$
(3) $$f(T) \in k[T] , x \in U \Longrightarrow x f(T) \in U $$$$L$$ is a submodule of $$M = K^n$$ if $$L$$ is a subset of $$K^n$$ such that:
(1) $$0 \in L$$
(2) $$x,y \in L \Longrightarrow x + y \in L
$$
(3) $$x \in L$$ and $$f(T) \in K[T] \Longrightarrow xf(T) \in L$$
===========================
Now assume that the subspace $$U$$ is a submodule $$L$$ of $$M$$
Then we have that $$xf(T) \in U$$ for all $$x \in L, f(T) \in K[T] $$
We need to show $$AU \subseteq U$$
So let $$x \in AU$$
Therefore
$$x = AU = \begin{pmatrix} a_{11} & a_{12} & a_{13} & ... & ... & a_{1n} \\ a_{21} & a_{22} & a_{23} & ... & ... & a_{2n} \\ a_{31} & a_{32} & a_{33} & ... & ... & a_{3n} \\... & ... & ... & ... & ... \\ ... & ... & ... & ... & ... \\ a_{n1} & a_{n2} & a_{n3} & ... & ... & a_{nn}\end{pmatrix} \begin{pmatrix} u_1 \\ u_2 \\u_3 \\.\\ .\\ u_s \\ 0 \\ 0 \\ .\\ . \\0 \end{pmatrix}$$ for some integer $$s$$ where $$s = dim(U)$$
Therefore
$$x = AU = \begin{pmatrix} a_{11}u_1 + a_{12}u_2 + a_{13}u_3 + ... \ ... + a_{1s}u_s + 0 +0 ... \ ... +0 \\ a_{21}u_1 + a_{22}u_2 + a_{23}u_3 + ... \ ... + a_{2s}u_s + 0 +0 ... \ ... +0 \\ a_{31}u_1 + a_{32}u_2 + a_{33}u_3 + ... ... + a_{3s}u_s + 0 +0 ... \ ... +0 \\... \ ... \\ ... \ ... \\ a_{n1}u_1 + a_{n2}u_2 + a_{n3}u_3 + ... ... + a_{ns}u_s +0 +0 ... \ ... +0 \end{pmatrix}$$
Thus $$x = AU \in U$$ if $$a_{ij}u_j \in U$$ since the terms of the matrix $$AU$$ are sums of such elements and a subspace will contain these sums if the individual summands belong to it ...
But we are given that $$U$$ is also a submodule L
Thus $$xf(T) \in L$$ ...
... ... I was going to try to use this to show that each term $$a_{ij}u_j \in U$$ ... ... BUT ... feel I have lost my way ...
Can someone please help ... ...
Peter
I need help with Exercise 1.2.8 (a) (Chapter 1: Basics, page 3o) concerning $$K^n$$ as a $$K[T]$$-module ... ...
First, so that MHB readers will understand the relevant notation for the construction of $$K^n$$ as a $$K[T]$$-module, I am presenting the relevant text from B&K as follows:https://www.physicsforums.com/attachments/2997
Exercise 1.2.8 (a) (page 30) reads as follows:https://www.physicsforums.com/attachments/2998
https://www.physicsforums.com/attachments/2999So we are given that:
$$A$$ is an $$n \times n$$ matrix over a field $$K$$
$$U$$ is a subspace of $$K^n$$
$$M$$ is the $$K[T]$$-module obtained from the vector space of column vectors $$K^n$$
So we regard $$M=K^n$$ as a right module over $$K[T]$$
Addition in $$M=K^n$$ is normal column vector addition $$x+y$$ where $$x, y \in K^n$$ making $$M=K^n$$ an abelian group as required
Given an $$n \times n$$ matrix $$A$$ over $$K$$, a right action of $$f(T) \in K[T]$$ on $$M$$ is defined as follows:
$$xf(T) = xf_0 + Axf_1 + ... \ ... A^rxf_r
$$
where
$$f(T) = f_0 + f_1T + ... \ ... f_rT^r$$
We are required to show that:
------------------------------------------------
... a subspace $$U$$ of $$K^n$$ is a submodule $$L$$ of $$M$$ ...
if and only if
... $$AU \subseteq U$$
------------------------------------------------
Just reviewing the definitions of subspace and submodule in this context ...$$U$$ is a subspace of $$K^n$$ if $$U$$ is a subset of $$K^n$$ such that:
(1) $$0 \in U$$
(2) $$x, y \in U \Longrightarrow x + y \in U $$
(3) $$f(T) \in k[T] , x \in U \Longrightarrow x f(T) \in U $$$$L$$ is a submodule of $$M = K^n$$ if $$L$$ is a subset of $$K^n$$ such that:
(1) $$0 \in L$$
(2) $$x,y \in L \Longrightarrow x + y \in L
$$
(3) $$x \in L$$ and $$f(T) \in K[T] \Longrightarrow xf(T) \in L$$
===========================
Now assume that the subspace $$U$$ is a submodule $$L$$ of $$M$$
Then we have that $$xf(T) \in U$$ for all $$x \in L, f(T) \in K[T] $$
We need to show $$AU \subseteq U$$
So let $$x \in AU$$
Therefore
$$x = AU = \begin{pmatrix} a_{11} & a_{12} & a_{13} & ... & ... & a_{1n} \\ a_{21} & a_{22} & a_{23} & ... & ... & a_{2n} \\ a_{31} & a_{32} & a_{33} & ... & ... & a_{3n} \\... & ... & ... & ... & ... \\ ... & ... & ... & ... & ... \\ a_{n1} & a_{n2} & a_{n3} & ... & ... & a_{nn}\end{pmatrix} \begin{pmatrix} u_1 \\ u_2 \\u_3 \\.\\ .\\ u_s \\ 0 \\ 0 \\ .\\ . \\0 \end{pmatrix}$$ for some integer $$s$$ where $$s = dim(U)$$
Therefore
$$x = AU = \begin{pmatrix} a_{11}u_1 + a_{12}u_2 + a_{13}u_3 + ... \ ... + a_{1s}u_s + 0 +0 ... \ ... +0 \\ a_{21}u_1 + a_{22}u_2 + a_{23}u_3 + ... \ ... + a_{2s}u_s + 0 +0 ... \ ... +0 \\ a_{31}u_1 + a_{32}u_2 + a_{33}u_3 + ... ... + a_{3s}u_s + 0 +0 ... \ ... +0 \\... \ ... \\ ... \ ... \\ a_{n1}u_1 + a_{n2}u_2 + a_{n3}u_3 + ... ... + a_{ns}u_s +0 +0 ... \ ... +0 \end{pmatrix}$$
Thus $$x = AU \in U$$ if $$a_{ij}u_j \in U$$ since the terms of the matrix $$AU$$ are sums of such elements and a subspace will contain these sums if the individual summands belong to it ...
But we are given that $$U$$ is also a submodule L
Thus $$xf(T) \in L$$ ...
... ... I was going to try to use this to show that each term $$a_{ij}u_j \in U$$ ... ... BUT ... feel I have lost my way ...
Can someone please help ... ...
Peter