Finding 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.

 

[HallsofIvy]

"v = a1U1 + a2U2 + ... + anUn "

There are two things wrong with this. First, you left out the information that U1= (1,1,1,1), U2= (0,1,1,1), U3= (0,0,11), and U3= (0,0,0,1).
Yes, you did tell us the basis but we (and you perhaps) needed to understand that the "U1", "U2", etc. that you were using WERE those vectors.

Second, you are using a1, a2, a3, a4 as the coefficients you want to find when you are already given a1, a2, a3, a4 as the components of the vector V! That' sure to confuse. Suppose we call the coefficients "c1", "c2", "c3", "c4" instead.

As hrc969 said, just replace U1, U2, U3, U4 in your equation by those basis vector, do the addition and set the components of the resulting vector equal to a1, a2, a3, a4. Solve the four equations for c1, c2, c3, and c4.