Define the kernel of a matrix.

The kernel of a matrix is the set of all vectors that are mapped to the zero vector.

In linear algebra, a matrix is a rectangular array of numbers arranged in rows and columns. The kernel of a matrix, also known as the null space, is the set of all vectors that are mapped to the zero vector when multiplied by the matrix. In other words, it is the set of solutions to the homogeneous equation Ax = 0, where A is the matrix and x is a vector.

To find the kernel of a matrix, we can use Gaussian elimination to row-reduce the augmented matrix [A|0]. The kernel is then the set of all solutions to the resulting system of linear equations. For example, consider the matrix A = [1 2 3; 4 5 6; 7 8 9]. We can row-reduce the augmented matrix as follows:

[1 2 3 | 0]
[4 5 6 | 0]
[7 8 9 | 0]

R2 - 4R1 -> R2
R3 - 7R1 -> R3

[1 2 3 | 0]
[0 -3 -6 | 0]
[0 -6 -12 | 0]

R3 - 2R2 -> R3

[1 2 3 | 0]
[0 -3 -6 | 0]
[0 0 0 | 0]

The last row represents the equation 0x1 + 0x2 + 0x3 = 0, which has infinitely many solutions. Therefore, the kernel of A is the set of all vectors of the form x = [-2s; s; t], where s and t are arbitrary constants.