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I am reading Paul E. Bland's book, "Rings and Their Modules".
I am trying to understand Section 2.2 on free modules and need help with the proof of Proposition 2.2.10.
Proposition 2.2.10 and its proof read as follows:View attachment 3587My question/problem is concerned with Bland's proof of Proposition 2.2.10 above.
Bland asks us to consider two bases $$\mathscr{B}$$ and $$\mathscr{B}'$$ where
$$\mathscr{B} = \{ x_1,x_2, \ ... \ ... \ x_n \} $$
and
$$ \mathscr{B}' = \{ y_1,y_2, \ ... \ ... \ y_m \} $$ Bland then goes through a process whereby he starts with the basis
$$\mathscr{B} = \{ x_1,x_2, \ ... \ ... \ x_n \} $$
and replaces various $$x_i$$ with $$y_i$$ until he reaches the basis
$$\{ \text{ the } x_i \text{ not eliminated } \} \cup \{ y_1,y_2, \ ... \ ... \ y_m \}$$THEN ... ... Bland argues as follows:
" ... ... But $$\mathscr{B}'$$ is a maximal set of linearly independent vectors of V, so it cannot be the case the $$n \lt m$$. Hence $$n \ge m$$. Interchanging $$\mathscr{B}$$ and $$\mathscr{B}'$$ in the argument gives $$m \ge n$$ and this completes the proof. ... ... "HOWEVER ... ... we know (straight away, without going through the construction process), that since $$\mathscr{B}$$ and $$\mathscr{B}'$$ are both maximal sets of linearly independent vectors that we cannot have $$n \lt m$$, nor can we have $$m \gt n$$, so we must have $$m = n$$.My question is as follows:
What is the relevance of the construction process above that results in the basis:
$$\{ \text{ the } x_i \text{ not eliminated } \} \cup \{ y_1,y_2, \ ... \ ... \ y_m \}$$?
Why do we need this process/construction?
Indeed, the argument for the Proposition seems to follow straight from the fact that both $$\mathscr{B}$$ and $$\mathscr{B}'$$ are both maximal sets of linearly independent vectors ... ... ?
Can someone please clarify the above issue?
Peter
I am trying to understand Section 2.2 on free modules and need help with the proof of Proposition 2.2.10.
Proposition 2.2.10 and its proof read as follows:View attachment 3587My question/problem is concerned with Bland's proof of Proposition 2.2.10 above.
Bland asks us to consider two bases $$\mathscr{B}$$ and $$\mathscr{B}'$$ where
$$\mathscr{B} = \{ x_1,x_2, \ ... \ ... \ x_n \} $$
and
$$ \mathscr{B}' = \{ y_1,y_2, \ ... \ ... \ y_m \} $$ Bland then goes through a process whereby he starts with the basis
$$\mathscr{B} = \{ x_1,x_2, \ ... \ ... \ x_n \} $$
and replaces various $$x_i$$ with $$y_i$$ until he reaches the basis
$$\{ \text{ the } x_i \text{ not eliminated } \} \cup \{ y_1,y_2, \ ... \ ... \ y_m \}$$THEN ... ... Bland argues as follows:
" ... ... But $$\mathscr{B}'$$ is a maximal set of linearly independent vectors of V, so it cannot be the case the $$n \lt m$$. Hence $$n \ge m$$. Interchanging $$\mathscr{B}$$ and $$\mathscr{B}'$$ in the argument gives $$m \ge n$$ and this completes the proof. ... ... "HOWEVER ... ... we know (straight away, without going through the construction process), that since $$\mathscr{B}$$ and $$\mathscr{B}'$$ are both maximal sets of linearly independent vectors that we cannot have $$n \lt m$$, nor can we have $$m \gt n$$, so we must have $$m = n$$.My question is as follows:
What is the relevance of the construction process above that results in the basis:
$$\{ \text{ the } x_i \text{ not eliminated } \} \cup \{ y_1,y_2, \ ... \ ... \ y_m \}$$?
Why do we need this process/construction?
Indeed, the argument for the Proposition seems to follow straight from the fact that both $$\mathscr{B}$$ and $$\mathscr{B}'$$ are both maximal sets of linearly independent vectors ... ... ?
Can someone please clarify the above issue?
Peter
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