Proving Linear Independence of vectors

[zecuria]

Homework Statement

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

Homework Equations

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

 

[ehild]

Homework Statement

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

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

ehild