Matrix of linear transformation

[gruba]

Homework Statement

Let $A:\mathbb R_2[x]\rightarrow \mathbb R_2[x]$ is a linear transformation defined as $(A(p))(x)=p'(x+1)$ where $\mathbb R_2[x]$ is the space of polynomials of the second order. Find all $a,b,c\in\mathbb R$ such that the matrix $\begin{bmatrix} a & 1 & 0 \\ b & 0 & 1 \\ c & 0 & 0 \\ \end{bmatrix}$ is the matrix of linear transformation $A$ with respect to some arbitrary basis of $\mathbb R_2[x]$.

Homework Equations

-Polynomial vector space
-Basis

The Attempt at a Solution

If we choose the standard basis, $$\mathcal B=\{1,x,x^2\}\Rightarrow p(x)=\alpha+\beta x+\gamma x^2,\alpha,\beta,\gamma\in\mathbb R\Rightarrow (A(p))(x)=\beta+(\beta+2\gamma)x+2\gamma x^2\Rightarrow$$

$A(1)=0x^2+0x+1,A(x)=0x^2+1x+1,A(x^2)=2x^2+0x+0$

Setting $A(1),A(x),A(x^2)$ as column vectors gives the matrix $\begin{bmatrix} 0 & 0 & 2 \\ 0 & 1 & 0 \\ 1 & 1 & 0 \\ \end{bmatrix}$ that is not of the form of given matrix $\begin{bmatrix} a & 1 & 0 \\ b & 0 & 1 \\ c & 0 & 0 \\ \end{bmatrix}$.

This means that we can't choose the standard basis to get matrix of $A$ that will be of the form $\begin{bmatrix} a & 1 & 0 \\ b & 0 & 1 \\ c & 0 & 0 \\ \end{bmatrix}$.

Question: Do we have to guess a proper basis? If not, then how to find one?

 

[micromass]

What is the relation between the matrix of a linear map in one basis and the matrix of the same linear map in another basis?

 

[micromass]

Alternatively, do you know a property of linear maps which is independent of the basis?

 

[gruba]

What is the relation between the matrix of a linear map in one basis and the matrix of the same linear map in another basis?

Let $B=\{b_1,b_2,b_3\}$ is a standard basis, and $B'=\{{b'}_1,{b'}_2,{b'}_3\}$.
Matrix of changing basis from $B$ to $B'$ is defined as $S=[[b_1]_{B'},...,[{b'}_n]_{B'}]$.

Finding $[b_1]_{B'},...,[{b'}_n]_{B'}$ gives matrix $S$.

This should be the reversed process, right? We know vectors $[b_1]_{B'},...,[{b'}_n]_{B'}$ in some basis,
and we need to find $b_1,...,b_n$ from basis $B$.

 

[micromass]

Right, but that is not my point. Let's say I give you two matrices $A$ and $B$ which are matrices of the same linear map but with different bases. Do you know anything about how $A$ and $B$ are related? Does similarity tell you something?

 

[gruba]

Right, but that is not my point. Let's say I give you two matrices $A$ and $B$ which are matrices of the same linear map but with different bases. Do you know anything about how $A$ and $B$ are related? Does similarity tell you something?

If $B$ is the matrix after transition to new basis, then $B=A^{-1}$?

 

[gruba]

@micromass The only condition is $c\neq 0$ since from $b_1=1,b_2=x,b_3=x^2$ follows ${b'}_1=x,{b'}_2=x^2,{b'}_3=\frac{1-ax-bx^2}{c}$.
Is this correct?

 

[gruba]

ATTEMPT EDITED:

Let $p(x)=a+bx+cx^2$ be a polynomial in standard basis $\mathcal B=\{1,x,x^2\}$ of $\mathbb R^2[x]$.
Then, linear transformation $A$ is defined as

$(A(p))(x)=p'(x+1)=(2+b)+2cx+0x^2$

From the given matrix, $\begin{bmatrix} a & 1 & 0 \\ b & 0 & 1 \\ c & 0 & 0 \\ \end{bmatrix}$

we have

$$[b_1]_{B'}= \begin{bmatrix} a \\ b \\ c \\ \end{bmatrix},[b_2]_{B'}= \begin{bmatrix} 1 \\ 0 \\ 0 \\ \end{bmatrix},[b_3]_{B'}= \begin{bmatrix} 0 \\ 1 \\ 0 \\ \end{bmatrix}$$

where $\mathcal {B'}$ is some basis different from standard basis $\mathcal B$.

Now, we have

$$b_1=a\cdot {b'}_1+b\cdot {b'}_2+c\cdot {b'}_3$$

$$b_2=1\cdot {b'}_1+0\cdot {b'}_2+0\cdot {b'}_3$$

$$b_3=0\cdot {b'}_1+1\cdot {b'}_2+0\cdot {b'}_3$$

where $b_1,b_2,b_3$ form standard basis $\mathcal B$, and
${b'}_1,{b'}_2,{b'}_3$ form new basis $\mathcal{B'}$.

From above equations,

$${b'}_1=x$$

$${b'}_2=x^2$$

$${b'}_3=\frac{1-ax-bx^2}{c},c\neq 0$$

and we have $\mathcal{B'}=\left\{x,x^2,\frac{1-ax-bx^2}{c}\right\}$.

Now we need to find the matrix of linear transformation $A$ with respect to basis $\mathcal{B'}$.

$$\begin{bmatrix} a & 1 & 0\\ b & 0 & 1\\ c & 0 & 0\\ \end{bmatrix} \begin{bmatrix} 2+b \\ 2c \\ 0 \\ \end{bmatrix} =\begin{bmatrix} 2a+ab+2c \\ 2b+b^2 \\ 2c+bc \\ \end{bmatrix}$$

Linear transformation $A$ in basis $\mathcal B'$ is given by

$$((A(p))(x))'=(2a+ab+2c)x+b(2+b)x^2+c(2+b)\frac{1-ax-bx^2}{c}$$

$$=(2a+ab+2c)x+b(2+b)x^2+(2+b)(1-ax-bx^2)$$

Conclusion: For all $a,b$ and for all $c\neq 0$, the matrix given by $\begin{bmatrix} a & 1 & 0 \\ b & 0 & 1 \\ c & 0 & 0 \\ \end{bmatrix}$ is the matrix of linear transformation $A$ in some basis $\mathcal B'$.

Is this correct?

 

[Mark44]

@gruba, this is a pretty strong hint:

Do you know anything about how AAA and BBB are related? Does similarity tell you something?

 

[gruba]

@gruba, this is a pretty strong hint:

So, is it correct that one is the inverse of the other?

 

[Mark44]

So, is it correct that one is the inverse of the other?

No.