Hi everyone, would greatly appreciate some help

Suppose that V is a vector space and U and W subspaces. Assume v= u (direct sum) w

S and T are basis for U and V respectively.

s = u1, u2,..., uk and t = w1, w2,...wt

Show that S U T is a basis for V (U= union)

I know I have to prove span and linear independence, and have done span, but it is linear independence that Im struggling with. I have tried writing out a relation containing all the elements of SUT with coefficients, and making this equal zero, but I cant see how this shows that the coefficients themselves must also equal zero...

Thanks in advance, tom.