linear algebra definitions

Question	Answer
linear subspace of a vector field	a linear subspace $W \subset V$ of a vector field $V$ is a subset that contains the zero vector, and for any $a,\,b \in W, a + b \in W$ and $\lambda a \in W$ for an arbitrary scalar $\lambda$ .
span	the span of $S \subseteq V$ is the set consisting of all linear combinations of the vectors in $S$ . a vector space is finite-dimensional if it is spanned by a finite set of vectors and infinite-dimensional otherwise.
linear map	a function $L: V \rightarrow U$ such that $L \left( v + w \right) = L \left(v\right) + L \left(u\right), \forall v, \, w \in V$ and $L \left( \lambda v \right) = \lambda L \left(v\right), \forall v \in V, \lambda \in \mathbb{F}$ . a linear ODE system is an example of a linear map. a matrix can be identified with a map
matrix representation of a linear map	let $V \subseteq \mathbb{R}^{m}$ with basis $\left\{ v_j \right\}_{j=1}^{m}$ and $U \subseteq \mathbb{R}^n$ with basis $\left\{u_i\right\}_{i=1}^{n}$ . the $n \times m$ matrix $P = \left[p_{ij}\right]$ that corresponds to the linear map $L: V \rightarrow U$ under the specified bases satisfies: $L \left(v_j\right) = p_{1j}u_1 + \ldots + p_{nj}u_n \; \forall j = 1,\ldots,m$

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	Created by Eric Andersen almost 10 years ago