Suppose that T Rn rightarrow Rm is a linear transformation

Suppose that T : R^n rightarrow R^m is a linear transformation, and suppose that A is the standard matrix T

Solution

The kernel (also known as null space or null space) of a linear map T : R^m Rⁿ between two vector spaces R^m and Rⁿ, is the set of all elements v of Rⁿ for which T(v) = 0, where 0 denotes the zero vector in Rⁿ. Consider a linear map represented as a m × n matrix A with coefficients in a field K (typically the field of the real numbers ) and operating on column vectors x with n components over K. The kernel of this linear map is the set of solutions to the equation A x = 0, where 0 is understood as the zero vector and A is the standard matrix for the linear map T. The dimension of the kernel of A is called the nullity of A. The column space of a matrix A is the span (set of all possible linear combinations) of its column vectors. The rank of a matrix is defined as the maximum number of linearly independent column vectors in the matrix which, in turn, is the column space of the matrix.

The columns of the standard matrix A of the linear transformation T are the images of e₁ , e₂, e₃, …, e_m where e₁ , e₂, e₃, …, e_m are the columns of the m x n identity matrix I_mxn .