374 - More Work for Examples in Section 8.6

More work for Example 1

A={{1,-2,2},{-2,1,-2},{2,-2,1}}
Out[]=
{{1,-2,2},{-2,1,-2},{2,-2,1}}
MatrixForm[A]
Out[]//MatrixForm=
1
-2
2
-2
1
-2
2
-2
1
Eigenvalues[A]
Out[]=
{5,-1,-1}
Eigenvectors[A]
Out[]=
{{1,-1,1},{-1,0,1},{1,1,0}}
Eigensystem[A]
Out[]=
{{5,-1,-1},{{1,-1,1},{-1,0,1},{1,1,0}}}
In[]:=
W[t_]:=Transpose[{{1,-1,1}
5t

,{-1,0,1}
-t

,{1,1,0}
-t

}]
MatrixForm[W[t]]
Out[]//MatrixForm=
5t

-
-t

-t

-
5t

0
-t

5t

-t

0
Det[W[t]]
Out[]=
-3
3t

Since det[W(t)] is non-zero for all t, the functions
X1(t)=
5t

-
5t

5t

,
X2(t)=
-
-t

0
-t

, and
X3(t)=
-t

-t

0
are linearly independent on (-∞,∞), by Theorem 8.5.

More work for Example 2

In[]:=
Clear[W]
In[]:=
W[t_]:=Transpose{3,1}
-3t

,{3,1}t+
1
2
,0
-3t


MatrixForm[W[t]]
Out[]//MatrixForm=
3
-3t

-3t

1
2
+3t
-3t

-3t

t
Det[W[t]]
Out[]=
-
1
2
-6t

Since det[W(t)] is non-zero for all t, the functions
X1(t)=
3
-3t

-3t

and
X2(t)=
-3t

1
2
+3t
-3t

t
are linearly independent on (-∞,∞)