How can you tell if a vector is in the null space (Linear Algebra)?

A vector y is in the null space of a matrix A if

Ay = 0

So check if multiplying the matrix by the vector is 0 (a zero vector).

Null(A) or N(A) is a quick hand for the Null area of A The Null area of A is the gap defined by ability of the vectors that are the suggestions of the equation Ax = 0 So if there are 2 self sustaining suggestions to Ax = 0 (no longer counting x = 0) then N(A) is 2d Null area is composed of 0 considering that's a answer. that's shown that N(A) is a ideal vector area considering its closed over addition and multiplication. n is the style of columns in A. that's an identical because of the fact the style of dimensions interior the the Column area of A, if the columns are linearly self sustaining. If the columns of A are self sustaining then dim(N(A)) = 0 In different words, Rank of A is the length of the column area of A. the version between the Rank of A and the style of columns is the length of the Null area of A. So, Rank(A) + Null (A) = n formulation wisely links the Rank(A) to the dimensions of the N(A) and the style of columns of A. notice, not one of the argument above assumes A is a sq. matrix

If a non-zero vector x is to be a candidate for the null space, simply find the matrix product Ax and see if it is 0.

If you want to discover the null space for a given matrix;

x is in the null space of A iff Ax = 0.

If A is non-singular, or invertible [detA not 0], the null space consists only of x = 0.

If A is singular, you can proceed to solve the homogeneous linear equations Ax = 0, for x(i)/x(n) where x(n) is assumed to be non-zero.