Linear system/Basis completion/2/Exercise

We consider the linear equations

${}8x-3y+5z+7u+6v=0\,,$

${}9x+2y+\,\,z\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,-v=0\,,$

${}\,\,\,\,\,\,\,\,\,\,\,\,\,\,7y-\,\,z+4u\,\,\,\,\,\,\,\,\,\,\,\,\,\,=0\,,$

over ${}\mathbb {R}$.

1. Determine a basis ${}{\mathfrak {b}}_{1}$ of the solution space of this linear system.
2. Extend the basis ${}{\mathfrak {b}}_{1}$ to a basis ${}{\mathfrak {b}}_{2}$ of the solution space of the linear system consisting of the first two equations.
3. Extend the basis ${}{\mathfrak {b}}_{2}$ to a basis ${}{\mathfrak {b}}_{3}$ of the solution space of the linear system consisting of the first equation alone.
4. Extend the basis ${}{\mathfrak {b}}_{3}$ to a basis ${}{\mathfrak {b}}_{4}$ of the total space ${}\mathbb {R} ^{5}$.

Create a solution