1. The vectors ${\mathbf{v}}_1$ and ${\mathbf{v}}_2$ form a basis for ${\mathbf{R}}^2$.

2. The vectors ${\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3$ do not form a basis for ${\mathbf{R}}^3$.

3. The given vectors do not form a basis for ${\mathbf{R}}^3$.

4. The given vectors do not form a basis for ${\mathbf{R}}^4$.

5. The three vectors ${\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3$ do not form a basis for ${\mathbf{R}}^3$.

6. The four given vectors form a basis for ${\mathbf{R}}^3$.

7. The three given vectors form a basis for ${\mathbf{R}}^3$.

8. The three given vectors form a basis for ${\mathbf{R}}^4$.

9. The plane $x-2y+5z=0$ is a 2-dimensional subspace of ${\mathbf{R}}^3$ with basis consisting of the vectors ${\mathbf{v}}_1=(2,1,0)$ and ${\mathbf{v}}_2=(-5,0,1)$.

10. The plane $y-z=0$ is a 2-dimensional subspace of ${\mathbf{R}}^3$ with basis consisting of the vectors ${\mathbf{v}}_1=(1,0,0)$ and ${\mathbf{v}}_2=(0,1,1)$.

11. The line is a 1-dimensional subspace of ${\mathbf{R}}^3$ with basis consisting of the vector $\mathbf{v}=(-3,1,1)$.

12. Hence the subspace consisting of all such vectors is 3-dimensional with basis consisting of the vectors ${\mathbf{v}}_1\text{‘}=(1,1,0,0)\text{,}{\mathbf{v}}_2=(1,0,1,0)$, and ${\mathbf{v}}_3=(1,0,0,1)$.

13. The subspace consisting of all such vectors is 2-dimensional with basis consisting of the vectors ${\mathbf{v}}_1=(3,0,1,0)$ and ${\mathbf{v}}_2=(0,4,0,1)$.

14. The subspace consisting of all such vectors is 2-dimensional with basis consisting of the vectors ${\mathbf{v}}_1=(-2,1,0,0)$ and ${\mathbf{v}}_2=(0,0,-3,1)$.

15. The solution space of the given system is 1-dimensional with basis consisting of the vector ${\mathbf{v}}_1=(11,7,1)$.

16. The solution space of the given system is 1-dimensional with basis consisting of the vector ${\mathbf{v}}_1=(11,-5,1)$.

17. The solution space of the given system is 2-dimensional with basis consisting of the vectors ${\mathbf{v}}_1=(-11,-3,1,0)$ and ${\mathbf{v}}_2(-11,-5,0,1)$.

18. The solution space of the given system is 2-dimensional with basis consisting of the vectors ${\mathbf{v}}_1=(-3,1,0,0)$ and ${\mathbf{v}}_2(-25,0,5,1)$.

19. The solution space of the given system is 2-dimensional with basis consisting of the vectors ${\mathbf{v}}_1=(3,-2,1,0)$ and ${\mathbf{v}}_2=(-4,-3,0,1)$.

20. The solution space of the given system is 2-dimensional with basis consisting of the vectors ${\mathbf{v}}_1=(1,-3,1,0)$ and ${\mathbf{v}}_2=(-2,1,0,1)$.

21. The solution space of the given system is 2-dimensional with basis consisting of the vectors ${\mathbf{v}}_1=(-1,-1,1,0)$ and ${\mathbf{v}}_2=(-5,-3,0,1)$.

22. The solution space of the given system is 2-dimensional with basis consisting of the vectors ${\mathbf{v}}_1=(2,1,0,0)$ and ${\mathbf{v}}_2=(-5,0,-7,1)$.

23. The solution space of the given system is 1-dimensional with basis consisting of the vector ${\mathbf{v}}_1=(2,-3,1,0)$.

24. The solution space of the given system is 3-dimensional with basis consisting of the vectors ${\mathbf{v}}_1=(-2,2,1,0,0)\text{,}{\mathbf{v}}_2=(-1,3,0,1,0)$, and ${\mathbf{v}}_3=(-3,-1,0,0,1)$.

25. The solution space of the given system is 3-dimensional with basis consisting of the vectors ${\mathbf{v}}_1=(-2,1,0,0,0)\text{,}{\mathbf{v}}_2=(2,0,1,1,0)$, and ${\mathbf{v}}_3=(-3,0,-4,0,1)$.

26. The solution space of the given system is 2-dimensional with basis consisting of the vectors ${\mathbf{v}}_1=(-2,1,2,1,0)$ and ${\mathbf{v}}_2=(3,-4,5,0,1)$.