Thread: Detemine whether vector are linearly dependent

The vectors are (1,-2,3) (-1,3,2) and (-1,10,5) . Question asks to find if they are linearly independent.

So I've put them into matrix form, as below

$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & { - 1} & { - 1} \\
{ - 2} & 3 & {10} \\
3 & 2 & 5 \\
\end{array}} \right]
$

now start gauss-jordon elimination.

new row2 = old row2 + 2*row1

$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & { - 1} & { - 1} \\
0 & 1 & 8 \\
3 & 2 & 5 \\
\end{array}} \right]
$

new row3 = old row3 - 3*row1

$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & { - 1} & { - 1} \\
0 & 1 & 8 \\
0 & 5 & 8 \\
\end{array}} \right]
$

new row3 = old row3 - 5&row2

$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & { - 1} & { - 1} \\
0 & 1 & 8 \\
0 & 0 & { - 32} \\
\end{array}} \right]
$

new row3 = old row3 divided by 32

$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & { - 1} & { - 1} \\
0 & 1 & 8 \\
0 & 0 & 1 \\
\end{array}} \right]
$

new row2 = old row2 - 8*row3

$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & { - 1} & { - 1} \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array}} \right]
$

new row1 = old row1 + row3

$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & { - 1} & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array}} \right]
$

new row1 = old row1 + row 2

$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array}} \right]
$

Firstly is my working correct, and secondly I would think they a linearly independent for the fact that there is only one 1 in each column am I correct in thinking this?

Hello,

Originally Posted by Craka

The vectors are (1,-2,3) (-1,3,2) and (-1,10,5) . Question asks to find if they are linearly independent.

So I've put them into matrix form, as below

$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & { - 1} & { - 1} \\
{ - 2} & 3 & {10} \\
3 & 2 & 5 \\
\end{array}} \right]
$

now start gauss-jordon elimination.

new row2 = old row2 + 2*row1

$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & { - 1} & { - 1} \\
0 & 1 & 8 \\
3 & 2 & 5 \\
\end{array}} \right]
$

new row3 = old row3 - 3*row1

$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & { - 1} & { - 1} \\
0 & 1 & 8 \\
0 & 5 & 8 \\
\end{array}} \right]
$

new row3 = old row3 - 5&row2

$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & { - 1} & { - 1} \\
0 & 1 & 8 \\
0 & 0 & { - 32} \\
\end{array}} \right]
$

new row3 = old row3 divided by 32

$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & { - 1} & { - 1} \\
0 & 1 & 8 \\
0 & 0 & 1 \\
\end{array}} \right]
$

new row2 = old row2 - 8*row3

$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & { - 1} & { - 1} \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array}} \right]
$

new row1 = old row1 + row3

$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & { - 1} & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array}} \right]
$

new row1 = old row1 + row 2

$\displaystyle
\left[ {\begin{array}{*{20}c}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \\
\end{array}} \right]
$

Firstly is my working correct, and secondly I would think they a linearly independent for the fact that there is only one 1 in each column am I correct in thinking this?

Everything is correct
You are correct in thinking this.
Because the columns of the final matrix are independent (they're known to form the canonical basis of $\displaystyle \mathbb{R}^3$


Another way of doing it was to calculate the determinant of the matrix
$\displaystyle \begin{pmatrix} 1 & -1 & -1 \\
-2 & 3 & 10 \\
3 & 2 & 5\end{pmatrix}$
(see the code, he's much easier)
If the determinant is not 0, then the vectors are linearly independent.

Thanks Moo,

It would seem the finding the determinant would be a little more straight forward than using row reduction. We haven't covered determinant in the course yet, however I know I have used them for using in Cramer's rule, is it possible to use determinants for matrices that are not a square matrix or for 4x4 matrices and above?

Originally Posted by Craka

Thanks Moo,

It would seem the finding the determinant would be a little more straight forward than using row reduction. We haven't covered determinant in the course yet, however I know I have used them for using in Cramer's rule, is it possible to use determinants for matrices that are not a square matrix or for 4x4 matrices and above?

Determinants are used for square matrices. I don't know if there is a way to define determinants for non-square matrices, but you don't have to care about it
It is possible to use determinantes for nxn matrices in general, so of course it works for 4x4. But above 3x3 (Sarrus's rule), it becomes more difficult.


As for using the determinant to prove that vectors are linearly independent... You can have an intuitive idea with that Gauss-Jordan elimination. There is indeed a step when you come to an upper triangular matrix. We know that in this case, the determinant is equal to the product of the diagonal. So if it's 0, it means that there is a 0 in the diagonal. Now try to see the situation. If there is a 0 in the diagonal of an upper triangular matrix, can you write it as a combination of other vectors ?
Looks like you can

Unfortunately, I can't provide a formal proof (I don't even know if it is a theorem, but I'm quite sure it works xD) I just wanted to give you an intuitive thing.
If you haven't been taught this, then stick to Gauss-Jordan elimination. It looks like you do it well



As a side note, once you get an upper-triangular matrix in the Gauss-Jordan elimination, you can stop at this step. Because with the 0's at the left, you can easily see that vectors (in rows) are independent or not.
I'm sorry, my explanations may look messy... I have never liked explaining linear algebra