Relation To The Row Space
Let A be an m by n matrix (i.e., A has m rows and n columns). The product of A and the n-dimensional vector x can be written in terms of the dot product of vectors as follows:
Here a1, ..., am denote the rows of the matrix A. It follows that x is in the null space of A if and only if x is orthogonal (or perpendicular) to each of the row vectors of A (because if the dot product of two vectors is equal to zero they are by definition orthogonal).
The row space of a matrix A is the span of the row vectors of A. By the above reasoning, the null space of A is the orthogonal complement to the row space. That is, a vector x lies in the null space of A if and only if it is perpendicular to every vector in the row space of A.
The dimension of the row space of A is called the rank of A, and the dimension of the null space of A is called the nullity of A. These quantities are related by the equation
The equation above is known as the rank–nullity theorem.
Read more about this topic: Kernel (matrix)
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