The Null space of a matrix is the set of vectors which post multiplying column wise or pre multiplying row wise produce the zero vector, and are not the trivial zero vector themselves.
The set of all Vectors basis coefficients * lambda = λ*v = the set of all vectors {v}.
Null space is – N(A) the set {v} ,of all vectors(column vectors), such that A*v =0 and v !=0.
Left-Null space is -N(Transpose(A)) the set {v} of all vectors(row vectors), such that Transpose(v)*(A)=0 and Transpose(v) !=0.
Only matrices that are linearly dependent by columns/rows have a null space, and a determinant of zero.
Linearly independent matrices do not have a null space and do not have a determinant of zero.
The Null space is orthogonal to C(A) and Left Null space is orthogonal to R(A):
The dot product of an orthogonal vector with the column space (any possible linear combination of all the columns scaled by any scalers) of matrix (A) is 0:
v⊥C(A)=0–> left null space –> A*Transpose(v) = Transpose(v) * A = 0
and vice versa for the R(A) and the null space.
The dimensionality of the C(A) and the left-null space combine all the ambient space, and vice versa for the R(A) and the null space.
