# Question #93fe7

Jun 26, 2017

See below.

#### Explanation:

Subspace characterization

If $\left\{{f}_{1} , {f}_{2}\right\} \in V \Rightarrow \alpha {f}_{1} + \beta {f}_{2} \in V$ with $\left\{\alpha , \beta\right\} \in \mathbb{R}$

1) If ${f}_{1} \left(x\right) \ne 0 , \forall x \in \left[- 1 , 1\right]$ and ${f}_{1} \in V$

calling ${f}_{2} = {f}_{1}$ and $\left\{\alpha = 1 , \beta = - 1\right\}$ we have

$\alpha {f}_{1} + \beta {f}_{2} = 0$ so $\alpha {f}_{1} + \beta {f}_{2} \notin V$

2) If ${f}_{1} \left(a\right) = {f}_{2} \left(a\right) = 0$ with $\left\{{f}_{1} , {f}_{2}\right\} \in V$ then

$\alpha {f}_{1} \left(a\right) + \beta {f}_{2} \left(a\right) = 0 \Rightarrow \alpha {f}_{1} \left(a\right) + \beta {f}_{2} \left(a\right) \in V$

Finally 1) is false and 2) is true.