A matrix is a rectangular array of numbers or symbols arranged in rows and columns. It is commonly used in mathematics, physics, computer science, and other fields to represent and manipulate data and equations.
The kernel of a matrix, also known as the null space, is the set of all vectors that when multiplied by the matrix result in the zero vector. In other words, it is the set of all solutions to the equation Ax = 0, where A is the given matrix. The dimension of the kernel is known as the nullity of the matrix.
The image of a matrix is the set of all possible outputs that can be obtained by multiplying the matrix by any valid input vector. It is also known as the column space of the matrix. To find the image, we can use row reduction techniques or the Rank-Nullity theorem, which states that the dimension of the image is equal to the rank of the matrix.
The rank of a matrix is the maximum number of linearly independent columns (or rows) in the matrix. It can be determined by performing row reduction on the matrix and counting the number of non-zero rows in the reduced row-echelon form. The rank is also equal to the number of pivot columns in the matrix or the dimension of the image.
The Rank-Nullity theorem states that the sum of the rank and nullity of a matrix is equal to the number of columns (or rows) in the matrix. In other words, the rank and nullity are complementary. This means that as the rank of a matrix increases, the nullity decreases and vice versa. The rank and nullity also provide important information about the solutions to a system of linear equations represented by the matrix.