# Computing Linear operator on the basis B.

• Aug 5th 2008, 06:23 PM
Juancd08
Computing Linear operator on the basis B.
Compute [T]_B and determine whether B is a basis consisting of eigenvectors of T.

a) V= P_2(R), T(a+bx+cx^2)= (-4a+2b-2c)-(7a+3b+7c)x +(7a+b+5c)x^2,

and a basis B= {x-x^2, -1+X^2, -1-x+x^2}

b)V=M_2x2(R), $\displaystyle \begin{pmatrix}\ a & b \\ c & d\end{pmatrix}\$

= $\displaystyle \begin{pmatrix}\-7a-4b+4c-4d & b\\-8a-4b+5c-4d & d\end{pmatrix}\$

And the basis B = {$\displaystyle \begin{pmatrix}\ 1 & 0 \\ 1 & 0\end{pmatrix}$ $\displaystyle \begin{pmatrix}\ -1 & 2 \\ 0 & 0\end{pmatrix}$ $\displaystyle \begin{pmatrix}\ 1 & 0 \\ 2 & 0\end{pmatrix}$ $\displaystyle \begin{pmatrix}\ -1 & 0 \\ 0 & 2\end{pmatrix}$}
• Aug 8th 2008, 12:50 AM
I am going to call the standard basis for $\displaystyle P_2(\mathbb{R})\quad$ E for no good reason
$\displaystyle [T]_B = [E\to B][T][B \to E]$
Where $\displaystyle [E \to B]$ is the matrix that converts from basis E to B. We can get the matrix $\displaystyle [B \to E]$ by using the basis vectors as columns. The matrix $\displaystyle [E\to B]$ is simply the inverse of $\displaystyle [B \to E]$.
B is a basis consisting of eigenvectors of T iff $\displaystyle [T]_B$ is diagonal.