Final answer:
To find the nullspace of a matrix A, perform row reduction to reduced row echelon form to identify pivot columns and free variables. Then, generate vectors by setting free variables to specific values and solving for the pivot variables. The collection of such vectors forms a basis for the Null(A).
Step-by-step explanation:
To find an explicit description of the Null(A) or nullspace of a matrix A, one must identify all vectors that result in the zero vector when multiplied by A. In other words, if Ax = 0, then x is in the nullspace of A. An explicit description of the nullspace involves finding a basis for this space, which is a set of vectors that spans the space without redundancy.
To proceed, we use Gaussian elimination or row reduction techniques on the matrix A to bring it to its reduced row echelon form. Once in this form, we identify the pivot columns and the free variables. For every free variable, we can set it to a particular value (such as 1) while setting all other free variables to 0, and solve for the pivot variables, outlining a specific vector in the nullspace. Repeating this process for each free variable, we can generate a system of vectors that span the nullspace.
For instance, if we have a matrix A from a 3-dimensional space with 1 free variable, we might find that setting this free variable, let's call it z, to 1 yields a vector in the nullspace such as [x, y, 1], where x and y are the solutions for the pivot variables in terms of z. If no free variables exist, the nullspace consists of only the null vector.