Can Bravais Lattices Be Expressed as a Combination of Primitive Vectors?

[Mythbusters]

I have three primitive vectors a1,a2,a3 for the body-centered cubic (bcc) Bravais can be chosen as

a1=ax
a2=ay
a3=(a/2)(x+y+z)

or, for instance, as

b1=(a/2)(y+z-x)
b2=(a/2)(z+x-y)
b3=(a/2)(x+y-z)

where x,y,z are unit vectors.

Now I should show that any vector of the form

R=n1a1+n2a2+n3a3
where n1,n2,n3 are integers

can be presented as

R=m1b1+m2b2+m3b3
where m1,m2,m3 are integers

Do anyone have an idea how I can do this?
Does it help me if I construct reciprocal lattice?

//
Mythbusters - "Well, here's your problem"

 

[tiny-tim]

Hi Mythbusters!

dunno wot a reciprockle lattice is …

but all you need to do is to express each a as a combination of bs

Hint: to get you started, what is b1 + b2 + b3 ?