Question

Prove that the given subset W is a subspace of
V , or show why it is not a subspace of V .

V=c∞(R) , W is the set of functions f⊆V for which
lim f(x) = 0
x→∞

Answer 1

i am going to make a quick guess at no, reasoning that subspace must include the zero vector, V does not

Answer 2

wait, why doesn't V include zero?! the zero function has infinitely many derivitives right?!

Answer 3

oops sorry, i meant W does not include the zero vector

Answer 4

why doesn't W include the zero function?!

Answer 5

oh lord, i should learn to read. i thought it said
\[\lim_{x\to 0}f(x)=\infty\] not the other way around. yes it includes the zero vector! ignore me

Answer 6

ahaha ok, you were scaring me for a sec! i thought i was just starting ot understand this stuff! ok, i'll ask you, for the same question then
if it is
f(4)=1 instead of the function above, what would the answer be?

Answer 7

answer to first question is yes, since
a) zero vector is in W
b) a linear combination of elements of Wis also in W
c) a scaler multiple of anything in W is in W

Answer 8

for the second question the answer is certainly no

Answer 9

the zero vector is not there

Answer 10

also if f(4)=1 then
\[af(4)=a\] not one, so scaler multiples are not there

Answer 11

ok great! thank you!

Answer 12

can i ask you one more by any chance?!

Answer 13

V = M2×2(R), W is the set of matrices whose square is the zero matrix (i.e., those
matrices A with A^2 the zero matrix).

Answer 14

could you please post this on a new thread?

Answer 15

yep