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,

<br /> \begin{pmatrix}<br /> 1 &amp; -1 &amp; 1 &amp; -2 &amp; | &amp; 0\\<br /> 2 &amp; -1 &amp; 1 &amp; -1 &amp; | &amp; 0\\<br /> -1 &amp; -1 &amp; 1 &amp; -4 &amp; | &amp; 0\\<br /> 1 &amp; -1 &amp; 0 &amp; -1 &amp; | &amp; 0<br /> \end{pmatrix}<br />

This is row reduced to,

<br /> \begin{pmatrix}<br /> 1 &amp; -1 &amp; 1 &amp; -2 &amp; | &amp; 0\\<br /> 0 &amp; 1 &amp; -1 &amp; 3 &amp; | &amp; 0\\<br /> 0 &amp; 0 &amp; -1 &amp; 1 &amp; | &amp; 0\\<br /> 0 &amp; 0 &amp; 0 &amp; 0 &amp; | &amp; 0<br /> \end{pmatrix}<br />

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