How Does the Linear Operator \(\phi\) Transform Matrices to Polynomials?

[gruba]

Homework Statement

Let \phi:M_{2,2}\mathbb{(R)}\rightarrow \mathcal{P_2} be a linear operator defined as: (\phi(A))(x)=tr(AB+BA)+tr(AB-BA)x+tr(A+A^T)x^2 where
B=<br /> \begin{bmatrix}<br /> 3 &amp; -2 \\<br /> 2 &amp; -2 \\<br /> \end{bmatrix}<br />

Find rank,defect and one basis of an image and kernel of linear operator \phi.

Homework Equations

-Linear transformations

The Attempt at a Solution

Could someone explain how to find matrix of linear operator \phi?
Also, is it necessary to know the matrix A?

 

[RUber]

Start by using A = I, the identity. The linear operator is defined by its effect on the basis elements.

 

[gruba]

Homework Statement

Let \phi:M_{2,2}\mathbb{(R)}\rightarrow \mathcal{P_2} be a linear operator defined as: (\phi(A))(x)=tr(AB+BA)+tr(AB-BA)x+tr(A+A^T)x^2 where
B=<br /> \begin{bmatrix}<br /> 3 &amp; -2 \\<br /> 2 &amp; -2 \\<br /> \end{bmatrix}<br />

Find rank,defect and one basis of an image and kernel of linear operator \phi.

Homework Equations

-Linear transformations

The Attempt at a Solution

Could someone explain how to find matrix of linear operator \phi?
Also, is it necessary to know the matrix A?

I have forgotten to mention that P_2 is a space of polynomials with degree not larger than 2.
Still, I don't understand why matrix A is not given.

 

[Samy_A]

I have forgotten to mention that P_2 is a space of polynomials with degree not larger than 2.
Still, I don't understand why matrix A is not given.

A can be any matrix in $M_{2,2}$.

Homework Statement

Let \phi:M_{2,2}\mathbb{(R)}\rightarrow \mathcal{P_2} be a linear operator defined as: (\phi(A))(x)=tr(AB+BA)+tr(AB-BA)x+tr(A+A^T)x^2 where
B=<br /> \begin{bmatrix}<br /> 3 &amp; -2 \\<br /> 2 &amp; -2 \\<br /> \end{bmatrix}<br />

Here they tell you how the linear operator $\phi$ transforms any matrix of $M_{2,2}$ into a polynomial.

As for a tip, see what @RUber suggested.

 

[Samy_A]

Sorry, double posting.

 

[Ray Vickson]

I have forgotten to mention that P_2 is a space of polynomials with degree not larger than 2.
Still, I don't understand why matrix A is not given.

The matrix $A$ is not given because it can be any 2 x 2 real matrix, and the definition of $\phi$ tells you how $A$ maps into a quadratic polynomial.

One fairly straightforward approach would be to take
A = \pmatrix{a &amp; b \\ c &amp; d}
and compute the polynomial $\phi(A)(x)$ explicitly.