Final answer:
To find the bases and dimensions for col A and nul A of a matrix, perform linear algebraic operations, reduce the matrix to echelon form, and solve Ax = 0. The number of vectors in each basis gives the dimension of that subspace.
Step-by-step explanation:
To find the bases for the column space (col A) and null space (nul A) of a matrix A, as well as the dimension of these subspaces and an echelon form of A, one must perform certain linear algebraic operations. The column space of A, denoted col A, consists of all linear combinations of the columns of A.
To find a basis for col A, you need to reduce the matrix to its echelon form and identify the pivot columns; these columns will form a basis for col A. The null space of A, denoted nul A, consists of all solutions to the homogeneous equation Ax = 0.
To find a basis for nul A, you must solve this equation, typically by reducing A to its reduced row echelon form and identifying the free variables. The dimensions of these subspaces are given by the number of vectors in each basis.
The echelon form of A is a partially simplified form where each leading entry of a row is to the right of the leading entry of the row above it, and all entries below a leading entry are zero.