- #1
DanielFaraday
- 87
- 0
Homework Statement
Let E={1, x, x2,x3} be the standard ordered basis for the space P3. Show that G={1+x,1-x,1-x2,1-x3} is also a basis for P3, and write the change of basis matrix S from G to E.
Homework Equations
The Attempt at a Solution
Here's what I got:
[tex]
S_E^G=\left(
\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1
\end{array}
\right)
[/tex]
Now, to prove that this is also a basis, I just need to show that it has an inverse, right?
Here's the problem. If the above is correct, then when you multiply it by G you should get E, right? After all, it is the "change of basis matrix S from G to E". However, this isn't the case:
[tex]
\left(
\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1
\end{array}
\right).\left(
\begin{array}{c}
1+x \\
1-x \\
1-x^2 \\
1-x^3
\end{array}
\right)=\left(
\begin{array}{c}
4-x^2-x^3 \\
2 x \\
-1+x^2 \\
-1+x^3
\end{array}
\right)
[/tex]
Am I doing something wrong, or am I just confused about what a change of basis matrix is supposed to do?