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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.