# Null subspace of a single vector

1. May 29, 2006

### vix_cse

Hi:

was wondering if somebody can help me with this I came across in a paper.

$e_k$ is a vector of $k$ $1's. M is a matrix of size n \times k. The author talks about projecting$M$onto the null space of$e_k$. This is what confuses me. Which$x\$ apart from the 0-vector solves e_kx=0.

any help will be appreciated.

2. May 29, 2006

### doodle

If k = 2, then x = [1 -1]^T solves e_2^T x = 0.

3. May 30, 2006

### HallsofIvy

Staff Emeritus
I was a bit confused by your reference to "the null space of ek". A vector does not have a "null space", a matrix (or linear transformation) has a "null space". But then, reading doodle's response, it became clear that what you really mean was "orthogonal complement": the set of all vectors whose dot product with ek is 0. The orthogonal complement of a vector in n dimensional space is an n-1 dimensional space.

It might help to think of the situation in 3 dimensions, with an x,y,z, Cartesian coordinate system. The vector e1, the unit vector pointing along the x-axis, has orthogonal complement equal to the yz-plane, the set of all vectors of the form (0,y,z).

Last edited: May 25, 2009
4. May 30, 2006

### Hurkyl

Staff Emeritus
It sounds more like e_k is a 1xk matrix of 1's.

5. May 30, 2006

### vix_cse

thanks all for your replies. yeah, it does make sense if i think of it as hallsofivy pointed out : the set of all vectors whose dot product with ek is 0.

this is how the author gets to the part in the paper

Z is a n-by-n projection matrix satisfying Z^2 = Z. e is the same as i mentioned.

define Z1 = Z - (1/n) e*transpose(e)

It is easy to see that Z1 represents the projection of the matrix Z onto the null subspace of e.

I couldnt figure out how it was so easy.

thanks.

6. May 25, 2009

### alice20006

I am still confused. As hallsofivy mentioned, the null subspace of e is a set of vectors. By stating that "Z1 represents the projection of the matrix Z onto the null subspace of e.", Z1 should be vectors. But Z1 is matrix. Can anyone explain this? Thanks.