# Unique vector representation?

quasar_4
unique vector representation??

## 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.

## 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.

## 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??

hrc969

## 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.

## 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.

## 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??
This is a problem about representing vectors with respect to different bases.

We have the standard representation for vectors in F^n: (a1,a2,a3,a4) or a1(1,0,0,0)+a2(0,1,0,0)+a3(0,0,1,0)+a4(0,0,0,1).

Now if we want to represent the same vector but in the given basis we need to find c1,c2,c3,c4 so that

c1(1,1,1,1)+c2(0,1,1,1)+c3(0,0,1,1)+c4(0,0,0,1)=(a1,a2,a3,a4)

c1 has to equal a1 otherwise you can't get a1 in the first coordinate so we need:

a1(1,1,1,1)+c2(0,1,1,1)+c3(0,0,1,1)+c4(0,0,0,1)=(a1,a2,a3,a4)

Now notice that only the first two vectors a1(1,1,1,1),c2(0,1,1,1) contribute to the second coordinate. Restricting our attention to what happens with the second coordinate we need that a1+c2=a2. So let c2=a2-a1. Now we have:

a1(1,1,1,1)+(a2-a1)(0,1,1,1)+c3(0,0,1,1)+c4(0,0,0,1)=(a1,a2,a3,a4)

Only the first three terms contribute to the third coordinate we need:
a1+(a2-a1)+c3=a3 so let c3=a3-a2.

I could do the last one as well but you should be able to see how to get it.