# Determining Linear Independence/Dependence using Determinants

• December 14th 2010, 10:30 PM
Lprdgecko
Determining Linear Independence/Dependence using Determinants
I'm doing a practice test for my Matrix Algebra exam. One of the questions says to "Use determinants to decide if the set of four vectors shown below is linearly dependent. Explain your reasoning."

(Sorry, not sure how to type out matrices. These are all 4x1 matrices)

v1 = [3]
.......[5]
.......[2]
.......[0]

v2= [0]
......[0]
......[1]
......[1]

v3 = [0]
.......[-2]
.......[-3]
.......[1]

v4 = [3]
.......[3]
.......[0]
.......[2]

I know how to determine linear dependence/independence, but not by using determinants. I can't find any examples in the notes.

I'm thinking I'd put them all in an augmented matrix and find the determinant of that, but how would I tell if it is linearly dependent?
• December 14th 2010, 10:55 PM
FernandoRevilla
Consider the matrix

$A=(v_1,v_2,v_3,v_4)$

then,
$x_1v_1+x_2v_2+x_3v_3+x_4v_4=0\Leftrightarrow A\begin{pmatrix}x_1\\x_2\\x_3\\x_4\end{pmatrix}=\b egin{pmatrix}0\\0\\0\\0\end{pmatrix}$

Now, use Rouche-Fröbenius theorem.

Fernando Revilla
• December 14th 2010, 11:30 PM
Lprdgecko
I'm not sure what the Rouche-Fröbenius theorem is, but, after looking at other sites as well, I took the determinant of the 4x4 matrix [v1 v2 v3 v4] which I got to be -6, which is not equal to 0, so it is linearly dependent? Right?
• December 14th 2010, 11:58 PM
FernandoRevilla
Quote:

Originally Posted by Lprdgecko
I'm not sure what the Rouche-Fröbenius theorem is, but, after looking at other sites as well, I took the determinant of the 4x4 matrix [v1 v2 v3 v4] which I got to be -6, which is not equal to 0, so it is linearly dependent? Right?

No, it is linearly independent. If $\det A\neq 0$ then, $A$ is an invertible matrix, so:

$\begin{pmatrix}x_1\\x_2\\x_3\\x_4\end{pmatrix}=A^{-1}\begin{pmatrix}0\\0\\0\\0\end{pmatrix}=\begin{pm atrix}0\\0\\0\\0\end{pmatrix}$

Fernando Revilla
• December 15th 2010, 12:17 AM
Lprdgecko
Woopsies, just realized that determinant should be 0. I realized what I did wrong.
• December 15th 2010, 12:32 AM
FernandoRevilla
Quote:

Originally Posted by Lprdgecko
Woopsies, just realized that determinant should be 0. I realized what I did wrong.

All right, if $\det A=0$ then, the family is linearly dependent.

Fernando Revilla