Notation
Axler writes 
. Axler never abbreviates bases, although if he did, the notation would look like 
where 
and 
.
Tao writes ![{\displaystyle [T]_{\beta }^{\beta '}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fbbd547c40415a54f677796d36aeb19d93db86b)
, where the "in" basis is a subscript and the "out" basis is a superscript. With vectors, ![{\displaystyle [v]_{\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/033f6f6aac970d293f11389de3800cb6c6009d45)
is a row vector and ![{\displaystyle [v]^{\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2a3f170434aa664297bf169249d0a78f2afc4299)
is a column vector. This means that we can write ![{\displaystyle [T]_{\beta }^{\beta '}[v]^{\beta }=[v]^{\beta '}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eae9317110d36ad3cff61a6e89c9f782e32f3a2e)
and think of the lower 
canceling out the upper .
Treil writes ![{\displaystyle [T]_{\beta '\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4f826ff5ec64e42b0e234efa274c5b854e80f40)
or sometimes ![{\displaystyle [T]_{\beta ',\beta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1981585421dc296b2d350ab2f378d5ce01e25fc5)
. With vectors, we can write ![{\displaystyle [T]_{\beta ',\beta }[v]_{\beta }=[v]_{\beta '}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a95882512f721d3d2c964348e0141cf87d070f4f)
so that the bases "balance".