Question

If W is a subspace of the vector space R^n and a and b are vectors such that a+b∈W and a-b∈W, then both a and b belong to W.
T/F?

Answer 1

what defines a subspace?

Answer 2

u explaining or asking me??

Answer 3

a+b∈W means a+b is a linear combination of the basis vector so W
similarly for a-b
\[a+b= \sum_{i}c_i x_i \\ a-b= \sum_{i}d_i x_i\]
adding we get
\[2a= \sum_{i}(c_i+d_i) x_i \]
or
\[a= \sum_{i}\frac{(c_i+d_i)}{2} x_i \]
which says a is a linear combination of the basis vectors of W, so

a∈W

you can use the same argument for b

Answer 4

**a+b∈W means a+b is a linear combination of the basis vectors of W

Answer 5

how should I do the b since I see b-b is 0?

Answer 6

subtract the two equations
a+b - (a-b)

Answer 7

so it's T! but the plu and minus sign don't matter?

Answer 8

no , the plus or minus signs do not matter.

Answer 9

thanks:)