Linear Algebra: Standing Notation

(i) Base. This setting extends Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity: its conventions and notation are in force.

(ii) Fields and vector spaces. A field $K$ is written with its operations and constants $+$, $\cdot$, $0$ and $1$, and a vector space $V$ over $K$ with its addition $+$ and its scalar multiplication $(\lambda,v)\mapsto\lambda v$. In $V$, the zero vector is written $0_{V}$, or $0$ where no confusion with the zero of $K$ can arise; the negative of $v$ is written $-v$, and the difference of $u$ and $v$ is written $u-v$. For $n\in\mathbb{N}_{0}$ and vectors $x_{1},\dots,x_{n}\in V$, the sum $\sum_{i=1}^{n}x_{i}$ is as in Span of a Finite Family; Spanning Families (i).

(iii) Subspaces. A subspace of $V$ is a linear subspace of $V$.

(iv) Linear maps. For vector spaces $V$ and $W$ over $K$, a linear map $T$ from $V$ to $W$ and $v\in V$, the value $T(v)$ is also written $Tv$. The sets $\mathcal{L}(V,W)$ and $\mathcal{L}(V)$ are as in Operations on Linear Maps (i); sums $S+T$, scalar multiples $\lambda T$, composites $ST$ and the identity $I_{V}$ are as in Operations on Linear Maps (ii), Operations on Linear Maps (iii), Operations on Linear Maps (iv) and Operations on Linear Maps (v); and $\ker T$ and $\operatorname{im}T$ are as in Kernel and Image of a Linear Map (i) and Kernel and Image of a Linear Map (ii).

(v) Span and dimension. For $n\in\mathbb{N}_{0}$ and $v_{1},\dots,v_{n}\in V$, $\operatorname{span}(v_{1},\dots,v_{n})$ is as in Span of a Finite Family; Spanning Families (ii); for a finite-dimensional vector space $V$, $\dim V$ is as in Finite-Dimensional Vector Spaces and Their Dimension (ii).