Homework Help: Proving Linear Independence of vectors

1. Jan 26, 2014

zecuria

1. The problem statement, all variables and given/known data

Let x1 = (1, 2, -1, 1), x2 = (-1, -1, -1, -1), x3 = (1, 1, 1, 0), x4 = (-2, -1 -4 -1)

Show that x1, x3 and x4 are linearly independent

2. Relevant equations

3. The attempt at a solution
Now I used the equation:
ax1+bx2+cx3+dx4=0

Hence forth the augmented matrix of the equation is,

$$\begin{pmatrix} 1 & -1 & 1 & -2 & | & 0\\ 2 & -1 & 1 & -1 & | & 0\\ -1 & -1 & 1 & -4 & | & 0\\ 1 & -1 & 0 & -1 & | & 0 \end{pmatrix}$$

This is row reduced to,

$$\begin{pmatrix} 1 & -1 & 1 & -2 & | & 0\\ 0 & 1 & -1 & 3 & | & 0\\ 0 & 0 & -1 & 1 & | & 0\\ 0 & 0 & 0 & 0 & | & 0 \end{pmatrix}$$

From as there is no leading entry corresponding to d, Setting d = t, the general solution is:

a = -t, b = -2t, c = t, d = t.

And as the number of leading entries =! number of unknowns so the vectors are linearly dependant

This is the point where I get confused as the question asks how they are linearly independant so I am quite confused at this point

Any help would be most appreciated and thanks in advanced

2. Jan 26, 2014

ehild

You have to show that x1,x2,x4 are linearly independent. Do the Gauss elimination with these three vectors.

ehild