Reducing a Matrix of Variables

[cal.queen92]

Homework Statement

Given the following vectors: (don't give values to variables)

V0 = (1 1 1 1) V1 = (x1 x2 x3 x4) V2 = (x1^2 x2^2 x3^2 x4^2)

V3 = (x1^3 x2^3 x3^3 x4^3)

a) prove that the 4x4 matrix A, whose columns are the vectors above, row reduces to the 4x4 identity matrix.

b) Do the Vectors above span IR^4? Are these vectors linearly independent?

Homework Equations

No equations, just have to know how to row reduce, etc...

If I'm not mistaken, an identity matrix is one in which the leading entries are all 1's and descend diagonally.

The Attempt at a Solution

It's hard to show via keyboard what I did, but I first created the matrix (so the first column is all ones, etc...). Then I row reduced to get rid of the last 3 ones of the first coloumn by the following operation:

(-1)R1 + R2 -- (-1) R1 + R3 -- (-1)R1 + R4

So now my first column has a leading one (in terms of rows) followed by all zeros below it. However, I'm left with a mix of variables in the other columns (like -x1 + x2, etc...) that I can't figure out how to get rid of! I keep going in circles.

Can anyone help me? I realize it's hard to describe this in words and can't attach a windows journal file for reference...

Thank you!

 

[HallsofIvy]

(Click on the formulas to see the LaTeX code used.)
So your matrix is
$$\begin{bmatrix}1 & x_1 & x_1^2 & x_1^3 \\ 1 & x_2 & x_2^2 & x_2^3 \\ 1 & x_3 & x_3^2 & x_3^3 \\ 1 & x_4 & x_4^2 & x_4^3\end{bmatrix}$$

Your first row reductions gives
$$\begin{bmatrix}1 & x_1 & x_1^2 & x_1^3 \\ 0 & x_2- x_1 & x_2^2- x_1^2 & x_2^3- x_1^3\\ 0 & x_3- x_1 & x_3^2-x_2^3 & x_3^3- x_1^3 \\ 0 & x_4- x_1 & x_4^2- x_1^2 & x_4^3- x_1^3\end{bmatrix}$$

Now the obvious next step is to divide the second row by $x_2- x_1$- and that is not as complicated as you might think. Remember that $x_2^2- x_1^2= (x_2- x_1)(x_2+x_1)$ and that $x_2^3- x_1^3= (x_2- x_1)(x_2^2+ x_2x_1+ x_1^3$.

 

[cal.queen92]

Great! thank you, I did that, now I have a leading one in the second row. I;ve tried the next step two ways,

1) eliminate second entry of first row
2) make all entries below leading one of second row 1's

But neither helps me move forward

 

[cal.queen92]

I also tried first obtaining the diagonal 1's that I need, but it gets too messy!