# Is there a list of facts that shows if a matrix is linearly (in)dependent?

• Dec 4th 2011, 08:42 PM
Reefer
Is there a list of facts that shows if a matrix is linearly (in)dependent?
Hi there!

I was wondering if there is a list of facts showing what makes a linearly independent and linearly dependent matrix?

For example, if there is a 3x3 matrix with a row of zeros, does that make the matrix linearly dependent?

For example, how is this matrix linearly independent?

[16 1]
[1 17]
[0 0]
[0 0]

Doesn't this have a row of zeros, so it should be dependent?

the matrix:
[1 3 -5]
[5 2 7]
[0 0 0]

Is linearly dependent because it has a row of zeros.
• Dec 4th 2011, 10:39 PM
FernandoRevilla
Re: Is there a list of facts that shows if a matrix is linearly (in)dependent?
Quote:

Originally Posted by Reefer
For example, if there is a 3x3 matrix with a row of zeros, does that make the matrix linearly dependent?

I'm afraid you are confusing concepts. If $\displaystyle V$ is a vector space and $\displaystyle v\in V$ then, $\displaystyle \{v\}$ is linearly independent if and only if $\displaystyle v\neq 0$ . So, a $\displaystyle 3\times 3$ matrix is linearly independent if an only if it is different from the zero matrix .

Quote:

For example, how is this matrix linearly independent?
[16 1]
[1 17]
[0 0]
[0 0]
Doesn't this have a row of zeros, so it should be dependent?
According to the above comments, it is linearly independent in $\displaystyle M_{4\times 2}(\mathbb{R})$. Another thing is that its four rows are linearly dependent in $\displaystyle \mathbb{R}^2$
• Dec 5th 2011, 05:57 AM
dwsmith
Re: Is there a list of facts that shows if a matrix is linearly (in)dependent?
Quote:

Originally Posted by Reefer
Hi there!

Doesn't this have a row of zeros, so it should be dependent?

the matrix:
[1 3 -5]
[5 2 7]
[0 0 0]

Is linearly dependent because it has a row of zeros.

Yes because the determinant is zero. Square matrices with a determinant of zero are dependent.
• Dec 5th 2011, 07:43 AM
FernandoRevilla
Re: Is there a list of facts that shows if a matrix is linearly (in)dependent?
Quote:

Originally Posted by dwsmith
Yes because the determinant is zero. Square matrices with a determinant of zero are dependent.

I suppose you mean that its rows (also its columns) are dependent.