Proving condition for spanning vectors I have highlighted the bit I don't understand, what does it mean by in the linear form? And didn't the proof already state that 4 lines above?



The linear form is as outlined in the last sentence of the first paragraph, namely the linear function of the entries of b: $l$(b)$ = \sum_{i=1}^n \alpha_i b_i$ corresponding to a zero row. 
The statement which is four lines above what you highlighted states that the coefficient of $b_i$ is $1$ at the start i.e., before we start the Gaussian elimination process (by switching rows, multiplying rows by scalars etc). The highlighted statement is saying that if the original $i$-th row (after at least one step in the Gaussian elimination process) is a zero row, it must have become a zero row by addition/s of multiples of one or more above rows. So the linear form on the right of the vertical line will be ($b_i$ + linear combination of the $b_j's$), where $j \neq i$. That is, $b_i$ plus a linear combination of the other entries of b. In this expression, the coefficient of $b_i$ is $1$. 
