Question

Please explain the terms row space, column space and nullspace (kernel).

Answer 1

Row space- The vector space generated by the rows of a matrix viewed as vectors. The row space of a matrix with real entries is a subspace generated by elements of , hence its dimension is at most equal to . It is equal to the dimension of the column space.

Column space- The column space (also called the range or image) of a matrix is the span (set of all possible linear combinations) of its column vectors. The column space of a matrix A = [a1 ...an], written as Col A is a set of all linear combinations of columns of A i.e. Col A = Span{a1,...,an}.

Nullspace- The null space of an m × n matrix A, written as Nul A, is the solution set of the equation Ax = 0. In set notation, NulA= {x:x? Rⁿ and Ax=0}.