Final answer:
The dimension of the row space of matrix A is 1, as is the dimension of the column space, because A has 2 zero rows in echelon form. The dimension of the null space of A is 7, found by applying the Rank-Nullity Theorem to the given matrix dimensions.
Step-by-step explanation:
If a matrix A is a 3 × 8 matrix and in its echelon form it has 2 zero rows, this gives us information about the rank and the nullity of the matrix. In this context, the rank of a matrix is the dimension of its row space or column space, and they are always equal. The number of non-zero rows in echelon form represents the rank of the matrix A. Since there are 2 zero rows, this means there is 1 non-zero row; therefore, the dimension of the row space of A, which is the rank, is 1. Because the row space and the column space have the same dimension, the dimension of the column space of A is also 1.
Next, we can apply the Rank-Nullity Theorem, which states that the rank of a matrix plus the nullity (the dimension of the null space) must equal the number of columns in the matrix. So, if we have a 3 × 8 matrix, the nullity n must satisfy: rank(A) + n = 8. Since the rank is 1, we get n + 1 = 8, so the dimension of the null space of A is n = 7.