In the book "Linear Algebra Done Right" ( Sheldon Axler ), the basis reduction algorithm ( Proof ) is as follows:
To practice mathematical writing, I tried to write the algorithm in a similar fashion using my own understanding in the following manner:
Let $ B = \{ v_1,...,v_n \} $ be an arbitrary spanning set of $ V $ . For all $ v_i \in B $, where $ 1 \leq i \leq n $, we'll check whether $ v_i $ is a linear combination of the other vectors in $ B $, if yes then we'll delete $ v_i $ from $ B $ else if not, keep $ v_i $ in $ B $.
Note: My proposal of the algorithm can be written logically as ( not fully formal, but it's mostly for intuition ): $\forall \text{spanning set $ B=\{ v_1,...,v_n \} $ } . \forall v_i \in B ( 1\leq i \leq n ) . P(v_i) $
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where I denote the statement " check whether $ v_i $ is a linear combination of the other vectors in $ B $, if yes then we'll delete $ v_i $ from $ B $ else if not, keep $ v_i $ in $ B $ " as $ P(v_i)$
Is my proposal of writing the algorithm correct? is it equivalent to the algorithm in the book?