# About rank one matrix

1. May 20, 2012

### applechu

Hi:
I see an principle about rank one matrice in the book, and it says
if u=(1,2,3), $\nu$t=[1 3 10], with Ax=0,
the equation $\nu$tx=0;
The problem is I see an example like following:
s1=[-3
1
0]

s2=[-10
0
1]
The nullspace contains all combination of s1 and s2. and produces the plane
x+3y+10z=0, perpendicular to row(1,3,10). And it lead to the result
Nullspace perpendicular to row space. I didn't know what the result means and
how its imply, could any one give me any instruct about that, thanks.

Last edited: May 20, 2012
2. May 20, 2012

### Number Nine

...go on.

3. May 21, 2012

### applechu

sorry, rewrite the s1, s2;

s1=\begin{pmatrix}
-3\\
1\\
0
\end{pmatrix}
and s2= \begin{pmatrix}
-10\\
0\\
1
\end{pmatrix}

4. May 22, 2012

### HallsofIvy

You defined u and $\nu^t$ but what is A and what is x?

The "nullspace" and "row space"of what matrix?

I didn't know what the result means and
how its imply, could any one give me any instruct about that, thanks.[/QUOTE]

5. May 22, 2012

### applechu

Hi:
A is a matrix, A=\begin{bmatrix}1 & 3 & 10\\2 & 6 & 20\\3 & 9 & 30\end{bmatrix}
u=(1,2,3) , Thanks

6. May 26, 2012

### algebrat

The trick is to row reduce, we get

\begin{bmatrix}1 & 3 & 10\\0 & 0 & 0\\0 & 0 & 0\end{bmatrix}

So, let's think about this. To get to the reduced matrix, we can say we multiplied on the left by elementary matrices, leaving A to multiply any vector x on the other side. If x is is in the null space, in other words, if it solves Ax=0, wel then if and only if it's still in the null space after we multiplied by the elementary matrices. Therefore:

The nullspace of A is the same as that for the reduced echelon form.

For the row space, it's a little different, but the row reduced echelon form is helpful again. Since we were multiplying on the left, we changed the image, we change the column space. So the column space of A is not the same as the column space of the row reduced echelon form. But, the elementary matrices preserve linear dependence and linear independence, so the relations between columns is preserved. Blah blah, but what this means is:

The pivot columns of row reduced echelon form tell which columns of A span the column space of A.

So since the pivot columns of row reduced are just the leftmost column, the column space of A is spanned by it's left most column, that is (1,2,3).

For the null space of the row reduced echelon form, clearly it's where x+3y+10z=0. Trying z=0 and y=1 gives (-3,1,0), while y=0 and z=1 gives (-10,0,1).

Other problems may be more complicated, and ask for different forms of answers, but the two underlined sentences show how the row reduced form gets you closer to answers, and with a little cleverness, you can handle other problems.