LAC Linear maps and associated subspaces

Section 2.5 Linear maps and associated subspaces

Linear algebra is the study of vector spaces and linear maps. Indeed we use linear maps to understand and classify vector spaces. Here we review some basic ideas to be expanded upon later.

Given two vector spaces \(V,W\) defined over the same field \(F,\) a linear map \(T:V\to W\) is function which preserves structure. Its definition says it takes “sums to the corresponding sums” (\(T(v+v') = T(v) + T(v')\)), and it takes “scalar multiples to the corresponding scalar multiples” (\(T(\lambda v) = \lambda T(v)\)).

Starting from the definition of a linear map, one proves by induction that a linear map takes linear combinations of vectors in the domain to the same linear combination of the corresponding vectors in the codomain. More precisely we have

Proposition 2.5.1. Linear maps preserve structure.

Let \(V,W\) be vector spaces over a field \(F\text{,}\) and \(T:V\to W\) a linear map. Then for every finite collection of vectors \(v_1, \dots, v_r \in V\text{,}\) and scalars \(a_1, \dots a_r \in F\) we have

\begin{equation} T(a_1v_1+ \cdots+a_rv_r) = a_1T(v_1) + \cdots + a_r T(v_r).\tag{2.5.1} \end{equation}

Remark 2.5.2.

What makes linear maps special is that even though they are functions from \(V\to W\text{,}\) to understand them we need not define them for every vector \(v \in V\text{.}\) It is enough to define them on a linearly independent spanning set for \(V\text{,}\) for once we have defined \(T(v_1), \dots, T(v_r)\text{,}\) we know the definition of \(T(v)\) for every \(v\in \Span\{v_1, \dots, v_r\}.\) Linear independence plays a crucial role here, but we take that up in the next section.

For now, we content ourselves with defining two subspaces associated to a linear map \(T:V\to W,\) the kernel or nullspace and the image.

These sets are familiar for the linear map \(T:F^n \to F^m\) given by \(T(x) = Ax\text{,}\) where \(A\) is any element of \(M_{m\times n}(F).\) The kernel of \(T\) is

\begin{equation*} \ker T = \{x \in F^n \mid T(x) = Ax = 0\}, \end{equation*}

the set of solutions to the homogeneous linear system of equations \(Ax=0.\) And the image,

\begin{equation*} \Im T = \{b\in F^m\mid Ax = b \text{ is solvable}\}. \end{equation*}

Of course for this linear transformation, \(T(x)= Ax\text{,}\) we know that \(\Im T = C(A),\) the column space of \(A.\)

Note 2.5.3.

One theorem you prove is that for a linear map \(T:V\to W,\) it is always the case that \(\ker T\) is a subspace of \(V,\) and \(\Im T\) is a subspace of \(W\text{.}\)

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