The basis for null space is a set of vectors that span the null space of a matrix. These vectors are linearly independent and can be used to represent all solutions to the homogeneous equation Ax = 0, where A is the original matrix.
The basis for row space is a set of vectors that span the row space of a matrix. These vectors are linearly independent and can be used to represent all possible linear combinations of the rows of the matrix.
The dimension of null space is determined by finding the number of linearly independent vectors in the basis for the null space. This can be done by reducing the matrix to reduced row echelon form and counting the number of free variables.
No, the dimension of row space cannot be greater than the number of rows in a matrix. This is because the row space is spanned by the rows of the matrix, and the maximum number of linearly independent rows in a matrix is equal to the number of rows in the matrix.
The dimension of null space and row space are related by the rank-nullity theorem, which states that the dimension of null space plus the dimension of row space is equal to the number of columns in the matrix. In other words, the dimension of null space and row space are complementary.