### 2.34 Find the rank and the bases of the Null space for a matrix A

Problem: Find the rank and nullities of the following matrices, and ﬁnd the bases of the range space and the Null space.

$$A=\begin {pmatrix} 2 & 3 & 3 & 4 \\ 0 & -1 & -2 & 2 \\ 0 & 0 & 0 & 1 \end {pmatrix}$$

 Mathematica mat={{2,3,3,4}, {0,-1,-2,2}, {0,0,0,1}}; {nRow,nCol} = Dimensions[mat]  3,4 Print["Rank (or dimension of the range space)=", MatrixRank[mat]]   Rank (or dimension of the range space)=3  Print["Dimension of the Null Space=", nCol-MatrixRank[mat]]   Dimension of the Null Space=1  Print["Basis for Null Space=",NullSpace[mat]]  Basis for Null Space={{3,-4,2,0}} 

 Matlab A=[2 3 3 4; 0 -1 -2 2; 0 0 0 1] [nRow,nCol]=size(A); r = rank(A); fprintf('A range space dimension=%d\n',r); fprintf('A null space dimension= %d\n',nCol-r); fprintf('Basic for null space of A ='); null(A,'r')'  A range space dimension=3 A null space dimension= 1 Basic for null space of A = ans = 1.5000 -2.0000 1.0000 0 

 Maple restart; A:=Matrix([[2,3,3,4],[0,-1,-2,2],[0,0,0,1]]); LinearAlgebra:-Rank(A); LinearAlgebra:-ColumnDimension(A)-LinearAlgebra:-Rank(A); LinearAlgebra:-NullSpace(A)  3 1 [3/2 -2 1 0]