§ 2

In a linear space $\mathcal{L}$ the vectors $a_{1}, a_{2}$ and $a_{3}$ are given with coordinate columns

in the basis $e_{1}, e_{2}, e_{3}, e_{4}$. Find a basis of their linear span $\mathcal{L}^{\prime }$.

Find a system of equations defining the subspace $\mathcal{L}^{\prime }$ from problem 1.

Find some subspace $\mathcal{M}^{\prime }$ which together with the subspace $\mathcal{L}^{\prime }$ from problem 1 satisfies the condition $\mathcal{L}=\mathcal{L}^{\prime } \oplus \mathcal{M}^{\prime }$.

In problem 1, $\mathcal{L}^{\prime }$ is the linear span of the vectors $a_{1}, a_{2}, a_{3}$ of the linear space $\mathcal{L}$ whose coordinate columns in the basis $e_{1}, e_{2}, e_{3}, e_{4}$ are $\| 1,2,3,4\|^{T}$, $\| 5,6,7,8\|^{T}$ and $\| 9,10,11,12\|^{T}$.

The subspace $\mathcal{L}^{\prime }$ is defined in problem 1: it is the linear span of the vectors $a_{1}, a_{2}, a_{3}$ of a linear space $\mathcal{L}$ whose coordinate columns in the basis $e_{1}, e_{2}, e_{3}, e_{4}$ are $\| 1,2,3,4\|^{T}$, $\| 5,6,7,8\|^{T}$ and $\| 9,10,11,12\|^{T}$. The subspace $\mathcal{L}^{\prime \prime }$ is spanned by the vectors $b_{1}$ and $b_{2}$ with coordinates 1, 1, 1, 2 and 2, 2, 2, 5. Find

In a four-dimensional space there are given