MA22020: Advanced linear algebra

1.4 Linear maps

Definitions. A map \(\phi \colon V\to W\) of vector spaces over \(\F \) is a linear map (or, in older books, linear transformation) if
\(\seteqnumber{0}{1.}{0}\)
\begin{align*} \phi (v+w)&=\phi (v)+\phi (w)\\ \phi (\lambda v)&=\lambda \phi (v), \end{align*} for all \(v,w\in V\), \(\lambda \in \F \).

The kernel of \(\phi \) is \(\ker \phi :=\set {v\in V\st \phi (v)=0}\leq V\).

The image of \(\phi \) is \(\im \phi :=\set {\phi (v)\st v\in V}\leq W\).

Definition. A linear map \(\phi :V\to W\) is a (linear) isomorphism if there is a linear map \(\psi :W\to V\) such that
\(\seteqnumber{0}{1.}{0}\)
\begin{equation*} \psi \circ \phi =\id _V,\qquad \phi \circ \psi =\id _W. \end{equation*}

If there is an isomorphism \(V\to W\), say that \(V\) and \(W\) are isomorphic and write \(V\cong W\).

Lemma 1.3 (Algebra 1B, lemma 1.2.3). \(\phi :V\to W\) is an isomorphism if and only if \(\phi \) is a linear bijection (and then \(\psi =\phi ^{-1}\)).

Notation. For vector spaces \(V,W\) over \(\F \), denote by \(L_{\F }(V,W)\) (or simply \(L(V,W)\)) the set \(\set {\phi :V\to W\st \text {$\phi $ is linear}}\) of linear maps from \(V\) to \(W\).

Theorem 1.4 (Linearity is a linear condition). \(L(V,W)\) is a vector space under pointwise addition and scalar multiplication. Thus
\(\seteqnumber{0}{1.}{0}\)
\begin{align*} (\phi +\psi )(v)&:=\phi (v)+\psi (v)\\ (\lambda \phi )(v)&:=\lambda \phi (v), \end{align*} for all \(\phi ,\psi \in L(V,W)\), \(v\in V\) and \(\lambda \in \F \).

Proposition 1.5 (Extension by linearity). Let \(V,W\) be vector spaces over \(\F \). Let \(\lst {v}1n\) be a basis of \(V\) and \(\lst {w}1n\) any vectors in \(W\).

Then there is a unique \(\phi \in L(V,W)\) such that
\(\seteqnumber{0}{1.}{0}\)
\begin{equation} \label {eq:4} \phi (v_i)=w_i,\qquad 1\leq i\leq n. \end{equation}

Theorem 1.6 (Rank-nullity). Let \(\phi :V\to W\) be linear with \(V\) finite-dimensional. Then
\(\seteqnumber{0}{1.}{1}\)
\begin{equation*} \dim \im \phi +\dim \ker \phi =\dim V. \end{equation*}

Proposition 1.7. Let \(\phi :V\to W\) be linear with \(V,W\) finite-dimensional vector spaces of the same dimension: \(\dim V=\dim W\).

Then the following are equivalent:
- (1) \(\phi \) is injective.
- (2) \(\phi \) is surjective.
- (3) \(\phi \) is an isomorphism.