# Mapping a matrix to the null space

• jaobyccdee
In summary, the conversation discusses trying to run a model in MATLAB and finding a 3 by 3 matrix L with all entries being one times a scalar, while knowing that DL=0 and L is mapped to the null space. The solution is a 1 by 3 matrix, but L can be extended to a 3 by 3 matrix by repeating the null space vector. It is also possible to adjust the value of L using an alpha constant. The conversation also discusses how to build such a matrix in MATLAB and the constraints that must be met.
jaobyccdee

## Homework Statement

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.

## Homework Equations

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

## 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?

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:
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?

## What is the null space of a matrix?

The null space of a matrix is the set of all vectors that, when multiplied by the matrix, result in a zero vector. In other words, it is the set of all solutions to the equation Ax = 0, where A is the given matrix.

## Why is mapping a matrix to the null space important?

Mapping a matrix to the null space is important because it allows us to find the solutions to systems of linear equations. It also helps us understand the linear dependence and independence of the columns of a matrix.

## How do you map a matrix to the null space?

To map a matrix to the null space, we first reduce the matrix to its reduced row echelon form. Then, the pivot columns correspond to the linearly independent columns of the matrix, while the non-pivot columns correspond to the free variables. The free variables can take on any value, and their corresponding columns form the basis for the null space.

## Can a matrix have a null space with more than one dimension?

Yes, a matrix can have a null space with more than one dimension. This means that there are multiple solutions to the equation Ax = 0, and the null space is a subspace of the vector space in which the matrix operates.

## How does the null space relate to the rank of a matrix?

The rank of a matrix is equal to the number of linearly independent columns of the matrix. The dimension of the null space is equal to the number of non-pivot columns in the reduced row echelon form of the matrix. Therefore, the rank of a matrix plus the dimension of its null space equals the total number of columns in the matrix.

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