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Answer 1 · 2017-06-26T19:03:24

See below.

Subspace characterization

If ${f_1, f_2} in V rArr alpha f_1 + beta f_2 in V$ with ${alpha, beta} in RR$

1) If $f_1(x) ne 0, forall x in [-1,1]$ and $f_1 in V$

calling $f_2 = f_1$ and ${alpha = 1, beta = -1}$ we have

$alpha f_1 + beta f_2=0$ so $alpha f_1 + beta f_2 !in V$

2) If $f_1(a) = f_2(a) = 0$ with ${f_1, f_2} in V$ then

$alpha f_1(a)+beta f_2(a) = 0 rArr alpha f_1(a)+beta f_2(a) in V$

Finally 1) is false and 2) is true.