Linear form/K^3/General/Solution tuple/Exercise

Let ${}K$ be a field and ${}a,b,c\in K$.

a) Show that the vectors

${}{\begin{pmatrix}b\\-a\\0\end{pmatrix}},\,{\begin{pmatrix}0\\c\\-b\end{pmatrix}},\,{\begin{pmatrix}c\\0\\-a\end{pmatrix}}\in K^{3}\,$

are solutions of the linear equation

${}ax+by+cz=0\,.$

b) Show that these three vectors are linearly independent.

c) Under what conditions generate these vectors the solution space of the equation?

d) Under what conditions generate the first two vectors the solution space of the equation?

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