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 9 years ago