[Math] Common symbols in linear algebra

linear algebranotation

I would like to clarify what symbols are the most commonly used for the following concepts in linear algebra:

  1. Linear transformations from $V$ to $U$, and the set of all transformations.

  2. A $m$-by-$n$ matrix over the field $\mathbb{F}$, and the set of all matrices.

  3. The matrix associated with a linear map $T:V \to U$.

  4. Linear independence

  5. Isomorphism, Canonical isomorphism. (For linear maps and vector spaces)

  6. Linear map associated with tensor. Tensor (rank 2) associated with linear map.

Best Answer

Notations I've used for these:

  1. $\mathcal L(U,V) = \{f:U \to V \mid f \text{ is linear}\}$. One may write $f \in \mathcal L(U,V)$. It also helps to write $\mathcal L(U):=\mathcal L(U,U)$.
  2. $\Bbb F^{m \times n}$, or $\mathcal M_{m \times n}(\Bbb F)$
  3. Note that often you'd want to specify a choice of basis here. In any case, I'd use $[T]$ to mean the matrix of $T$, and $[T]_{\mathcal A \to \mathcal B}$ to denote the matrix of $T$ with respect to the bases $\mathcal A,\mathcal B$.
  4. No common symbols that I know of
  5. $U \cong V$ or $U\sim V$ is used to indicate that the spaces are isomorphic. Sometimes $f:U \overset{\sim}{\to}V$ is used to indicate that $f$ is an isomorphism between the spaces. I don't know of a special symbol for canonical isomorphism.
  6. Not sure what you mean here. Perhaps you mean the map $T \otimes T$.
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