Exercise 5.4.14

True or false? Explain.

1. Every set of $5$ vectors in $\mathbb{R}^{5}$ is linearly independent.

2. Every set of $4$ vectors in $\mathbb{R}^{5}$ is linearly independent.

3. Every set of $6$ vectors in $\mathbb{R}^{5}$ is linearly dependent.

4. No set of $4$ vectors spans $\mathbb{R}^{5}$.

5. Every linearly independent set of $5$ vectors in $\mathbb{R}^{5}$ is a basis of $\mathbb{R}^{5}$.

6. Every linearly independent set of $4$ vectors in $\mathbb{R}^{5}$ is a basis of $\mathbb{R}^{5}$.

7. Some linearly independent set of $4$ vectors in $\mathbb{R}^{5}$ is a basis of $\mathbb{R}^{5}$.

8. Every spanning set of $6$ vectors in $\mathbb{R}^{5}$ is a basis of $\mathbb{R}^{5}$.

9. Every linearly independent set of $4$ vectors in $\mathbb{R}^{5}$ spans a $4$-dimensional subspace of $\mathbb{R}^{5}$.