# Mapping a matrix to the null space

1. Mar 19, 2012

### jaobyccdee

1. The problem statement, all variables and given/known data
I am trying to run a model in matlab. D is a 2 by 3 matrix, Knowing that DL=0, which means L is mapped to the null space.

2. Relevant equations
How can i find L so that it is a 3 by 3 matrix with all its entries being one times a scalar.

3. The attempt at a solution
I used null(D) to find L, the solution is a 1X3 matrix (a vector). Since i want a three by three matrix, can i just say that L=[null(D), null(D), null(D)] since L is mapped to the null space(a zeros vector/matrix) thus does not matter if i extend it from a vector to a matrix anyways. Also, can i put an alpha in front of the L to ajust its value so that all its entries equal to 1?

2. Mar 19, 2012

### lanedance

you may need to give some more info on what you are trying to accomplish

geomtrically, we can consider the multiplication as follows;
$$D.L = \begin{pmatrix} d_1^T \\ d_2^T \\ \end{pmatrix} \begin{pmatrix} l_1 & l_2 & l_3 \end{pmatrix}$$

So as the column vectors l_i are perpindcular to all the row vectors d_j, you will have the required matrix.

Note that if d_1 and d_2 are linearly indpendent, then there will only be one unique l_i, upto a multiplicative constant

Last edited: Mar 19, 2012
3. Mar 19, 2012

### jaobyccdee

D is not linearly independent, and L has to be all one times a scalar. How should i ask matlab to build such a matrix?

4. Mar 19, 2012