Linear Dependence/Independence

[mexilie]

Homework Statement

Is the set S = {(3,-2,4),(4,6,-4),(3,6,-2),(-13,2,-18)} linearly dependent? If so, give a dependency equation.

Homework Equations

The Attempt at a Solution

I first placed the set S into a matrix equation
[3 4 3 -13
-2 6 6 2
4 -4 -2 -18]

then put it into rref:

[1 0 0 5.44 0
0 1 0 7.76 0
0 0 1 -9.24 0]

Since the final column is all zeros, is the system linearly independent?

 

[radou]

Can a set of more than 3 vectors in R^3 be linearly independent?

 

[mexilie]

If the row rank is less than the number of variables then there has to be a parametric solution. How would I write a dependency equation, then?

 

[Mark44]

This matrix,
[3 4 3 -13]
[-2 6 6 2]
[4 -4 -2 -18]
represents the vector equation c1*v1 + c2*v2 + c3*v3 + c4*v4 = 0. As a matrix equation this is Ac = 0, where the columns of A are your four vectors, and c = <c1, c2, c3, c4>^T.

Assuming that your work is correct and that you ended with the next matrix (I removed the 5th column of 0s),

[1 0 0 5.44]
[0 1 0 7.76]
[0 0 1 -9.24]

this matrix says that c1 + 5.44*c4 = 0, c2 + 7.76*c4 = 0, and c3 - 9.24*c4 = 0.