# Write down the matrix of this transformation with respect to the given ?basis

Let T : M2(R) → M2(R) be the linear transformation defined by T(A) = 2A + 3A^T. Write down the matrix of this transformation with respect to the basis {Ei, 1 ≤ i ≤ 4} where

                       E1 =    |1 0| ,
|0 0|

E2= |0 1|
|0 0|

E3 = |0 0| and
|1 0|
E4 =| 0 0|
| 0 1|


My attempt ;

T(|1 0|) = 2 |1 0| + 3 |1 0| = | 5 0|

|0 0|     |0 0|   |0 0|  | 0 0|

T (| 0 1| ) = 2 | 0 1| + 3  | 0 1| = | 0 5|
| 0 0|       | 0 0|      | 0 0|   | 0 0|


I Don't know whethear im in the rigt way or in wrong way....PLiz help me,

• $T(A)$ is only defined when $A$ is a $2\times2$ matrix. Therefore $T(1\ 0)$ does not really make sense. On the other hand $$T\left(\pmatrix{1&0\cr0&0\cr}\right)$$ et cetera will be relevant. – Jyrki Lahtonen Sep 7 '17 at 12:10
• Your $T$ matrix is $4\times4$. – kimchi lover Sep 7 '17 at 12:11
• im not getting @ jyrki Lahtonen – user469754 Sep 7 '17 at 12:13
• Please use MathJax Your calculations, at its present form, is hard to understand. You will get more respones if the question is easy to read and understand. – Krish Sep 7 '17 at 14:16
• i don't know how to use maths jax,i was trying last time also i write the symbol for example \int_{a}^{b} f(x) dx.....but it doesnot convert into mathematical form ,@ krish.. – user469754 Sep 7 '17 at 19:17

I assume that the question is to let $$T : M_{2 \times 2}(\Bbb R) \rightarrow M_{2 \times 2}(\Bbb R)$$ be the linear transformation $$T(A) = 2A + 3A^t,$$ and that we want to find the matrix of $$T.$$
Because the domain and co-domain each have four dimensions, the matrix of $$T$$ is a $$4 \times 4$$ matrix. The $$i$$th column of the matrix of $$T$$ is the coordinate matrix of $$T(E_i)$$ denoted by $$[T(E_i)].$$ We compute \begin{align} [T(E_1)] & = \left[ \begin{bmatrix}5 & 0\\ 0 & 0\end{bmatrix} \right] = [5E_1 + 0E_2 + 0E_3 + 0E_4] = \begin{bmatrix} 5\\ 0\\ 0\\ 0\end{bmatrix},\\\\ [T(E_2)] & = \left[ \begin{bmatrix}0 & 2\\ 3 & 0\end{bmatrix} \right] = [0E_1 + 2E_2 + 3E_3 + 0E_4] = \begin{bmatrix} 0\\ 2\\ 3\\ 0\end{bmatrix},\\\\ [T(E_3)] & = \left[ \begin{bmatrix}0 & 3\\ 2 & 0\end{bmatrix} \right] = [0E_1 + 3E_2 + 2E_3 + 0E_4] = \begin{bmatrix} 0\\ 3\\ 2\\ 0\end{bmatrix}, \mbox{ and}\\\\ [T(E_4)] & = \left[ \begin{bmatrix}0 & 0\\ 0 & 5\end{bmatrix} \right] = [0E_1 + 0E_2 + 0E_3 + 5E_4] = \begin{bmatrix} 0\\ 0\\ 0\\ 5\end{bmatrix}. \end{align} Thus, the matrix of $$T$$ is $$\begin{bmatrix} 5 & 0 & 0 & 0\\ 0 & 2 & 3 & 0\\ 0 & 3 & 2 & 0\\ 0 & 0 & 0 & 5\end{bmatrix}.$$