Find the matrix representation of T relative to the basis B = {1, t, t^2}

[Nemanjavuk69]

Hello everyone

I have the following problem I am working on, please open up the attached picture file.



I know the answer to this problem is [imath]M = \begin{bmatrix} 2 & 0 & 0\\ 0 & 3 & 4\\ 5 & 0 & -6 \end{bmatrix}[/imath]

However, I have no clue how this is being calculated, I would appreciate if someone could give me a detailed and easy way of understanding how to find the matrix representation of T relative to the basis [imath]B = \begin{Bmatrix} 1, & t, & t^2\\ \end{Bmatrix}[/imath]

 

[Steven G]

If you looked closely you will see a pattern!
Look at the result for T(a₀+a₁t+a₂t²)
How many a₀ do you have for the constant?
How many a₀ do you have for t?
How many a₀ do you have for t²?

 

[topsquark]

Hello everyone

I have the following problem I am working on, please open up the attached picture file.

View attachment 34728

I know the answer to this problem is [imath]M = \begin{bmatrix} 2 & 0 & 0\\ 0 & 3 & 4\\ 5 & 0 & -6 \end{bmatrix}[/imath]

However, I have no clue how this is being calculated, I would appreciate if someone could give me a detailed and easy way of understanding how to find the matrix representation of T relative to the basis [imath]B = \begin{Bmatrix} 1, & t, & t^2\\ \end{Bmatrix}[/imath]

Hint:

Say that we have an operator
[imath]A: \mathbb{P}_2 \to \mathbb{P}_2: \left ( \begin{matrix} a_0 \\ a_1 \\ a_2 \end{matrix} \right ) \mapsto \left ( \begin{matrix} a_0 \\ 0 \\ 0 \end{matrix} \right )[/imath].

Can you tell me what the matrix representation of A will be?

What if we had
[imath]B: \mathbb{P}_2 \to \mathbb{P}_2: \left ( \begin{matrix} a_0 \\ a_1 \\ a_2 \end{matrix} \right ) \mapsto \left ( \begin{matrix} a_0 \\ 0 \\ a_1 \end{matrix} \right )[/imath].

-Dan

 

[Steven G]

Multiply M by (a₀+a₁t+a₂t²)^T

Then all should be obvious.

 

[pka]

Hello everyone

I have the following problem I am working on, please open up the attached picture file.

View attachment 34728

I know the answer to this problem is [imath]M = \begin{bmatrix} 2 & 0 & 0\\ 0 & 3 & 4\\ 5 & 0 & -6 \end{bmatrix}[/imath]

However, I have no clue how this is being calculated, I would appreciate if someone could give me a detailed and easy way of understanding how to find the matrix representation of T relative to the basis [imath]B = \begin{Bmatrix} 1, & t, & t^2\\ \end{Bmatrix}[/imath]

To try an answer, one is tempted to ask if "you have studied any of this"?
The question gives the basis [imath]\mathcal{B}=\left\{1,t,t^2\right\}\text{ for }\mathbb{P}_2[/imath]
Moreover, the mapping [imath]\mathit{T}\left(a_0+a_1t+a_2t^2\right)=2a_0+\left(3a_1+4a_2\right)t+\left(5a_0-6a_2\right)t^2 [/imath]
clearly that gives us the transformation matrix: [imath] M=\begin{bmatrix} 2& 0 & 0 \\ 0 & 3 & 4 \\ 5 & 0 & -6\end{bmatrix} [/imath]

[imath][/imath][imath][/imath]