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**unique vector representation??**

**1. Homework Statement**

So, here I am again with another...

The problem gives me a basis {(1,1,1,1),(0,1,1,1),(0,0,1,1),(0,0,0,1)} in F^4. I am supposed to find the unique representation of an arbitrary vector (a1,a2,a3,a4) as a linear combination of the vectors in the basis.

**2. Homework Equations**

v = a1U1 + a2U2 + ... + anUn where v is the vector formed by the linear combination of vectors (U1,...,Un) and scalar coefficients (a1,...,an).

Each vector v determines a unique n-tuple of scalars (a1,...,an) and vice-versa.

**3. The Attempt at a Solution**

It seems to me that all one needs do is put these in the typical unique form

v= a1U1 + a2U2 + a3U3 + a4U4.

But the solution given in our book is given as:

v = a1U1 + (a2 - a1)U2 + (a3 - a2)U3 + (a4 - a3)U4.

I just can't seem to see where the (a2 - a1), (a3 - a2), and (a4 - a3) come from... any explanations anyone??