Example 5
Example 5
Clear[a,b,ab]
a={{1,2,-3},{3,6,-9},{-2,-4,6}};
b={{1},{0},{-3}};
MatrixForm[a]
MatrixForm[b]
Out[21]=
1 2 -3 3 6 -9 -2 -4 6
Out[22]=
1 0 -3
In[23]:=
ab=AppendRows[a,b];
MatrixForm[ab]
Out[23]=
1 2 -3 1 3 6 -9 0 -2 -4 6 -3
In[24]:=
MatrixForm[RowReduce[ab]]
Out[24]=
1 2 -3 0 0 0 0 1 0 0 0 0
In[25]:=
Clear[a,b,ab]
a={{1,2,-3},{3,6,-9},{-2,-4,6}};
b={{0},{0},{0}};
MatrixForm[a]
MatrixForm[b]
Out[25]=
1 2 -3 3 6 -9 -2 -4 6
Out[26]=
0 0 0
In[27]:=
ab=AppendRows[a,b];
MatrixForm[ab]
Out[27]=
1 2 -3 0 3 6 -9 0 -2 -4 6 0
In[28]:=
MatrixForm[RowReduce[ab]]
Out[28]=
1 2 -3 0 0 0 0 0 0 0 0 0
Before moving on to investigate some theoretical properties of matrices and the subspaces associated with them, let me tell you of some terminology that is sometimes used.
Variables that correspond to pivot columns in the GJSF of [A,b] are called basic variables, and the remaining ones are called nonbasic variables.
In the very first example, all the variables (x1, x2 ,x3) were basic.
In Example 2, x1, x2, and x3 were basic variables, and x4 was nonbasic.