To determine the dimension of the null space of matrix A, we can use the Rank-Nullity Theorem. This theorem states that for any matrix, the sum of the rank and the dimension of the null space is equal to the number of columns in the matrix.
In this case, matrix A is a 5×6 matrix, which means it has 6 columns. The rank of the matrix, which is the number of pivot columns, is given as 4. Therefore, the dimension of the null space (dim Nul A) can be calculated as follows:
dim Nul A = Number of columns – Rank
dim Nul A = 6 – 4 = 2
So the dimension of the null space is 2. This is true regardless of the fact that the matrix A has 5 rows but 6 columns. The reason we refer to R^4 in the question is a bit misleading; while A has dimensions of 5 rows, we calculate the null space based on the number of columns and the rank, which are independent of the dimension of the vector space the columns are drawn from.
In conclusion, dim Nul A=2 because the difference between the number of columns and the rank of A is 2, and it does not relate to it being in R^4.