Question

Describe various spaces associated with an m × n matrix A, such as null space, row space. column space and eigenspace. What are the relationships among them? How does the concept of a linear transformation and its properties relate to matrices and those spaces of the matrices?

Answer 1

If A is your matrix, the null-space is simply put, the set of all vectors v such that A ⋅ v = 0 . It's good to think of the matrix as a linear transformation; if you let h ( v ) = A ⋅ v , then the null-space is again the set of all vectors that are sent to the zero vector by h.

Let A be an m by n matrix. The space spanned by the rows of A is called the row space of A, denoted RS(A); it is a subspace of R ⁿ . The space spanned by the columns of A is called the column space of A, denoted CS(A); it is a subspace of R ^m .

The eigenspace corresponding to λ is by definition the solution space of A , and hence it always contains the zero vector. The eigenvector/s corresponding to λ are the non-zero vectors from the corresponding eigenspace.

It can be just as instructive to look at functions that are not linear transformations. Since the defining conditions must be true for all vectors and scalars, it is enough to find just one situation where the properties fail.