# linear maps

• Mar 9th 2008, 06:16 AM
linear maps
Let T be the linear map on $P_2$ defined by
$T(p)(t) = (1+t)p^{'}(t)$
Determine the matrix of T with respect to the basis $(1, t, t^2)$ of $P_2$.
What are the eigenvalues of T? Is T diagonalisable?
• Mar 9th 2008, 09:23 AM
Plato
I will help with the basic matrix.
From $T(a + bt + ct^2 ) = (1 + t)(b + 2ct) = \left( {b + (b + 2c)t + 2ct^2 } \right)$ we get the matrix
$\left[ {\begin{array}{ccc}
0 & 1 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 2 \\ \end{array}} \right]$
• Mar 9th 2008, 10:50 AM
CaptainBlack
Quote:

Let T be the linear map on $P_2$ defined by
$T(p)(t) = (1+t)p^{'}(t)$
Determine the matrix of T with respect to the basis $(1, t, t^2)$ of $P_2$.
What are the eigenvalues of T? Is T diagonalisable?

The columns of the matrix are the images of each of the basis vectors under
the transformation (in component form with respect to the basis)

RonL
• Mar 9th 2008, 11:11 AM
cheers guys. One more question, what does the symbol that looks like a "U to the power 'an upside down T'" mean? I dont know how to notate that with latex.
• Mar 9th 2008, 11:58 AM
Chloroform
lol, you miss those lectures too? I'm doing the same course.

The 'U to the power 'an upside down T' thing you speak of is the orthogonal complement of U. The orthogonal compliment of a subspace U of an inner product V is the set of all vectors in V that are orthogonal to every vector in U.

Know Q3 yet? :p
• Mar 9th 2008, 12:09 PM
no ive not really looked at it yet, Q.1s sorted. Trying to do the Stats assignment as well, bloody hard...
• Mar 9th 2008, 06:39 PM
lllll
For $T(p)(t) = (1+t)p^{'}(t)$

is $T$ invertible?

& would $T^{-1} = -b - (b+2c)t - (2c)t^2$ ?
• Mar 9th 2008, 09:01 PM
CaptainBlack
Quote:

Originally Posted by lllll
For $T(p)(t) = (1+t)p^{'}(t)$

is $T$ invertible?

& would $T^{-1} = -b - (b+2c)t - (2c)t^2$ ?

No $T$ is not invertible, the matrix of $T$ has zero determinant.

Another way to look at it is that if it were invertibable then for every quadratic function $f(x)$ (quadratic function here means a polynomial of degree not more than 2):

$p(t)=\int_0^t \frac{f(\zeta)}{1+\zeta} ~d\zeta + p_0$

would be a unique quadratic, but it is neither unique nor always a quadratic
(put $f(x)=1$ to see that it is not always a quadratic)

RonL