Fnd the left null space of matrix

A = [5 -3 1]
[-2 4 -6]
[11 -8 5]

To find the left null space of matrix A, we need to solve the equation A^T * x = 0, where A^T is the transpose of matrix A and x is the vector representing the left null space.

The transpose of matrix A is obtained by interchanging the rows and columns of A:
A^T = [5 -2 11]
[-3 4 -8]
[1 -6 5]

We want to find the null space of A^T, so we write the equation A^T * x = 0 in augmented matrix form:
[5 -2 11 | 0]
[-3 4 -8 | 0]
[1 -6 5 | 0]

To solve the augmented matrix, we perform row operations to reduce it to row-echelon form. Starting with the first column, we perform row operations to make the first entry (pivot) in the first row equal to 1 by dividing the first row by the value of the pivot:
[1 -0.4 2.2 | 0]
[0 0.2 -0.6 | 0]
[0 0 0 | 0]

Next, we perform row operations on the second column to make the second entry (pivot) in the second row equal to 1:
[1 -0.4 2.2 | 0]
[0 1 -3 | 0]
[0 0 0 | 0]

Now, we can see that the third column does not contain any pivots, which means it is a free variable. Let's denote it as t.

From the second row, we have:
1 * x1 - 3 * x2 = 0 => x1 = 3 * x2

So, the vector x can be represented as:
x = [3 * x2, x2, t]

Thus, the left null space of matrix A is the set of all vectors of the form [3 * x2, x2, t], where x2 and t can take any real values.