Verify the assertions from Example 2.18:

1. A list $v$ of one vector $v \in V$ is linearly independent if and only if $v \ne 0$.
2. A list of two vectors in $V$ is linearly independent if and only if neither is a scalar multiple of the other
3. $(1,0,0,0),(0,1,0,0),(0,0,1,0)$ is linearly independent in $\mathbf F^4$
4. The list $1,z,\dots,z^m$ is linearly independent in $\mathcal P(\mathbf F)$ for each nonnegative integer $m$

1. If $v = 0$ then $\lambda v = 0$ so not independent, if $v \ne 0$ then $\lambda v \ne 0$ for all $\lambda \ne 0$, so independent.
2. If $v_1,v_2 \in V$ is dependent then 2.2.1 (a) implies one is a scalar multiple of the other. If $v_1 = \lambda v_2$ then they're clearly dependent as $v_1 - \lambda v_2 = 0$.
3. Obvious as their sum is $(a_1,a_2,a_3,0)$ which is only zero when $a_1=a_2=a_3=0$.
4. We will show independence by showing that 2.21 (a) does not apply. Suppose

take the jth derivative to get $j! = 0$, contradicting their equality. and since 2.21 does not apply they cannot be dependent.