kolbergkd

2022-09-29

nick1337

To represent the transpose homomorphism from the vector space of 2x2 matrices over the field of real numbers, denoted as ${M}_{2×2}\left(ℝ\right)$, to itself, we need to consider the standard bases for both the domain and the codomain.
The standard basis for ${M}_{2×2}\left(ℝ\right)$ is given by:
${B}_{V}=\left\{\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]\right\}$.
Let's denote the transpose homomorphism as $A$. For a matrix $M$ in ${M}_{2×2}\left(ℝ\right)$, the transpose of $M$ is denoted as ${M}^{T}$. The transpose homomorphism $A$ can be defined as $A\left(M\right)={M}^{T}$.
Now, let's consider a matrix $M$ in ${M}_{2×2}\left(ℝ\right)$ represented in the standard basis ${B}_{V}$:
$M=a\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]+b\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]+c\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right]+d\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$,
where $a,b,c,d$ are real numbers.
The representation of $M$ with respect to the standard basis ${B}_{W}$ can be obtained by taking the transpose of $M$ and representing it as a linear combination of the basis vectors in ${B}_{W}$.
Since ${B}_{W}$ is the same as ${B}_{V}$ in this case, the representation of $M$ with respect to ${B}_{W}$ is the same as the original representation of $M$.
The representation of $M$ with respect to ${B}_{W}$ is given as:
$\left[M{\right]}_{{B}_{W}}=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$.
Therefore, the transpose homomorphism $A$ maps a matrix $M$ in ${M}_{2×2}\left(ℝ\right)$ to its transpose ${M}^{T}$ without changing the representation of $M$ with respect to the standard basis ${B}_{V}$.
The transpose homomorphism can be represented as:
$A:{M}_{2×2}\left(ℝ\right)\to {M}_{2×2}\left(ℝ\right)$, $A\left(M\right)={M}^{T}$.
And the representation of $M$ with respect to ${B}_{W}$ can be represented as:
$\left[M{\right]}_{{B}_{W}}=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]$.

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