MA20216: Algebra 2A

1.4 Linear maps

Definitions. A map \(\phi :V\to W\) of vector spaces over \(\F \) is a linear map (or, in older books, linear transformation) if
\(\seteqnumber{0}{1.}{1}\)
\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
\(\seteqnumber{0}{1.}{1}\)
\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 1.5 (Algebra 1B, lemma 1.2.3). \(\phi :V\to W\) is an isomorphism if and only if \(\phi \) is a linear bijection (and then \(\psi =\phi ^{-1}\)).

1.4.1 Vector spaces of linear maps

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 1.6 (Linearity is a linear condition). \(L(V,W)\) is a vector space under pointwise addition and scalar multiplication. Otherwise said, \(L(V,W)\leq W^V\).

1.4.2 Linear maps and matrices

Definition. Let \(V,W\) be finite-dimensional vector spaces over \(\F \) with bases \(\cB :\lst {v}1n\) and \(\cB ':\lst {w}1m\) respectively. Let \(\phi \in L(V,W)\). The matrix of \(\phi \) with respect to \(\cB ,\cB '\) is the matrix \(A=(A_{ij})\in M_{m\times n}(\F )\) defined by:
\(\seteqnumber{0}{1.}{1}\)
\begin{equation} \label {eq:27} \phi (v_j)=\sum _{i=1}^mA_{ij}w_{i}, \end{equation}

for all \(\bw 1jn\).

In the special case where \(V=W\) and \(\cB =\cB '\), we call \(A\) the matrix of \(\phi \) with respect to \(\cB \).

1.4.3 Extension by linearity

Proposition 1.7 (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
\(\seteqnumber{0}{1.}{2}\)
\begin{equation} \label {eq:2} \phi (v_i)=w_i,\qquad 1\leq i\leq n. \end{equation}

1.4.4 The rank-nullity theorem

Theorem 1.8 (Rank-nullity). Let \(\phi :V\to W\) be linear with \(V\) finite-dimensional. Then
\(\seteqnumber{0}{1.}{3}\)
\begin{equation*} \dim \im \phi +\dim \ker \phi =\dim V. \end{equation*}

Proposition 1.9. 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.