How to find a null space vector in Julia

Note: The null space can also be considered the set of all vectors perpendicular to the row space of matrix $A$. This is because if a vector is perpendicular to all the rows of $A$, then it will be orthogonal to any linear combination of those rows, which means it will be in the null space of $A$.

Example

Let matrix $A$ be

$$A=\left(\begin{array}{ccc} A_{01} & \ldots & A_{0 n} \\ \ldots & \ldots & \ldots \\ \ldots & \ldots & \ldots \\ A_{m 0} & \ldots & A_{m n} \end{array}\right)$$

and the vector $x$ be

$$x=\left(\begin{array}{c} x_0 \\ \cdot \\ \cdot \\ x_n \end{array}\right).$$

For $x$ to be the null space vector, the following equations should be satisfied:

$$A_{0 0} x_0+A_{0 1} x_1+\ldots \ldots + A_{0 n} x_n=0 \\ \cdot \\ \cdot \\ A_{m 0} x_0+A_{m 1} x_1+\ldots \ldots + A_{m n} x_n=0$$

Code example

Let’s see an example of finding the null space vector:

Code explanation

Here is a line-by-line explanation of the code above:

• Line 1: We import the LinearAlgebra module.

• Lines 3: We define a matrix $A$.

• Line 5: We find the nullspace vector of matrix $A$ using the nullspace function from the LinearAlgebra module.

• Line 6: We print the nullspace vector $x$ returned by the function.

• Line 8: We check if the vector $x$ satisfies the equation $Ax=0$.