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For an endless set of solutions of a system of linear equations, one possible representation is a specific solution plus any of possible solutions of the related homogeneous system. My prof and tutor suggest the following notation:

$$ \mathbb{L} = \begin{bmatrix} a \\ b \end{bmatrix} + \mathrm{span}\left\{\begin{bmatrix} c \\ d\end{bmatrix}\right\}\!\!, $$ where $\begin{bmatrix} a \\ b \end{bmatrix}$ and $\begin{bmatrix} c \\ d\end{bmatrix}$ are the particular and homogeneous solutions, respectively.

But I have a slight gut feeling that it's not really nice to add a vector to a set hoping to get another set in the end, so wouldn't it be nicer to write something like $$ \mathbb{L} = \left\{\begin{bmatrix} a \\ b \end{bmatrix} + \lambda\begin{bmatrix} c \\ d\end{bmatrix}\; \middle|\; \lambda \in \mathbb{R}\right\}? $$ Somehow they have banned this method while correcting my homework, so I'm a little stunned. Need to mention, this is a less complex linear algebra course for engineers, not mathematicians.

Are there, maybe, any other possible and rigorous ways to build the intended set and which of the ways above is, in the end, better?

P.S.: ($\mathbb{L}$ stands for the German "Lösung")

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It's not uncommon to write an element plus a set, and interpret it the way you have. However, I'm shocked that the notation you suggested would be "banned", as it is correct and very common notation.

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