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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 (c) ...
Exercise 1.2.8 (c) reads as follows:View attachment 5097Now ... from this exercise we have that $$M \equiv \mathbb{C}^3$$
... and ...
$$A = \begin{pmatrix} 0&1&1 \\ 0&0&1 \\ 0&0&0 \end{pmatrix}$$
Now ... given a vector $$m \in M \equiv \mathbb{C}^3$$ and the matrix $$A$$ we have that the products $$Am, A^2 M , \ ... \ ...$$ are all elements of $$\mathbb{C}^3$$ ...
So ... following B&K Example 1.2.2 (iv) ... ... see below ... ... consider $$M \equiv \mathbb{C}^3$$ as a right module over the polynomial ring $$\mathbb{C} [T]$$ where
$$m f(T) = mf_0 + Am f_1 + \ ... \ ... \ A^r f_r $$
where
$$f(T) = f_0 + f_1 T + \ ... \ ... \ f_r T^r \in \mathbb{C} [T]
$$
Now, we are given:
$$L(v) \equiv$$ submodule of M generated by vThat is $$L(v) \equiv v f(T)$$ ...
and
$$\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$$ ...
BUT ... now how do we show $$\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \in v f(T)$$ for some choice of $$f$$ ...
Can someone please help ... ?
Further ... can someone show me how to find all $$v$$ with dim$$(L(v)) = 2$$?Hope someone can help ...
Peter
I need help with Exercise 1.2.8 (c) ...
Exercise 1.2.8 (c) reads as follows:View attachment 5097Now ... from this exercise we have that $$M \equiv \mathbb{C}^3$$
... and ...
$$A = \begin{pmatrix} 0&1&1 \\ 0&0&1 \\ 0&0&0 \end{pmatrix}$$
Now ... given a vector $$m \in M \equiv \mathbb{C}^3$$ and the matrix $$A$$ we have that the products $$Am, A^2 M , \ ... \ ...$$ are all elements of $$\mathbb{C}^3$$ ...
So ... following B&K Example 1.2.2 (iv) ... ... see below ... ... consider $$M \equiv \mathbb{C}^3$$ as a right module over the polynomial ring $$\mathbb{C} [T]$$ where
$$m f(T) = mf_0 + Am f_1 + \ ... \ ... \ A^r f_r $$
where
$$f(T) = f_0 + f_1 T + \ ... \ ... \ f_r T^r \in \mathbb{C} [T]
$$
Now, we are given:
$$L(v) \equiv$$ submodule of M generated by vThat is $$L(v) \equiv v f(T)$$ ...
and
$$\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$$ ...
BUT ... now how do we show $$\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \in v f(T)$$ for some choice of $$f$$ ...
Can someone please help ... ?
Further ... can someone show me how to find all $$v$$ with dim$$(L(v)) = 2$$?Hope someone can help ...
Peter