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.

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 $\displaystyle \mathbb{R}^2$

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.

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.