Exercise 3.3.2

See Example 2 in this section.

1.

The set $\{(-3,1)\}$ is a basis and the dimension is $1$.

2.

The set $\{(\frac{1}{3},\frac{2}{3},1)\}$ is a basis and the dimension is $1$.

3.

The set $\{(-1,1,1)\}$ is a basis and the dimension is $1$.

4.

The set $\{(0,1,1)\}$ is a basis and the dimension is $1$.

5.

The set $\{(-2,1,0,0),(3,0,1,0),(-1,0,0,1)\}$ is a basis and the dimension is $3$.

6.

The set $\{(0,0)\}$ is a basis and the dimension is $0$.

7.

The set $\{(-3,1,1,0),(1,-1,0,1)\}$ is a basis and the dimension is $1$.