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
$B= \begin{bmatrix} 3 & -2 \\ 2 & -2 \\ \end{bmatrix}$

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
$B= \begin{bmatrix} 3 & -2 \\ 2 & -2 \\ \end{bmatrix}$

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
$B= \begin{bmatrix} 3 & -2 \\ 2 & -2 \\ \end{bmatrix}$

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
$$A = \pmatrix{a & b \\ c & d}$$
and compute the polynomial $\phi(A)(x)$ explicitly.