1

Does anybody know what the following notation means? (I'm referring to the subscript & superscript to the right of the square brackets)

$\psi(T)=[T]_β^γ$

In this case, $T: V \rightarrow W$ is a linear map on finite-dimensional vector spaces $V$ & $W$ (over the same field $F$) with dimensions $n$ and $m$ resp. And, $\beta$ & $\gamma$ are ordered bases for resp. $V$ & $W$. Lastly, $\psi$ is the isomorphism $\psi:L(V,W)→M_{m\times n}(F)$.

(Full context here Proving isomorphism between linear maps and matrices)

Josh
  • 31
  • 1
    Isn't the OP explaining it at the link:" But this means that $[T]_{\beta}^{\gamma} = A$"? – Dietrich Burde Feb 24 '21 at 16:20
  • I suspect it concerns the representation of linear transformations $T$ with respect to a fixed pair of ordered bases. – hardmath Feb 24 '21 at 16:23
  • If $\beta = (v_1,\dots,v_n)$ and $\gamma = (w_1,\dots,w_m)$, then $[T]{\beta}^\gamma$ is the $m \times n$ matrix with entries $a{ij}$ such that $T(v_j) = \sum_{i=1}^n a_{ij}w_i$ for all $j$. – azif00 Feb 24 '21 at 20:33

1 Answers1

1

The expression on the right represents the matrix of the operator $T:V\to W$ with respect to the bases $\beta$ of $V$ and $\gamma$ of $W$.

More specifically, one evaluates T at the $j$-th basis vector in $\beta$ and expresses the output as a linear combination of basis vectors from $\gamma$. The respective coefficients in the linear combination form the $j$-th column of the matrix $[T]_{\beta}^{\gamma}$.

Anon
  • 46