Chapter 1 Linear algebra: concepts and examples

Let us warm up by revising some of the key ideas from Algebra 1B. Along the way, we will see some new examples and prove a couple of new results.

1.1 Vector spaces

Recall from Algebra 1B, §1.1:

  • Definition. A vector space \(V\) over a field \(\F \) is a set \(V\) with two operations:

    addition

    \(V\times V\to V: (v,w)\mapsto v+w\) with respect to which \(V\) is an abelian group:

    • • \(v+w=w+v\), for all \(v,w\in V\);

    • • \(u+(v+w)=(u+v)+w\), for all \(u,v,w\in V\);

    • • there is a zero element \(0\in V\) for which \(v+0=v=0+v\), for all \(v\in V\);

    • • each element \(v\in V\) has an additive inverse \(-v\in V\) for which \(v+(-v)=0=(-v)+v\).

    scalar multiplication

    \(\F \times V\to V: (\lambda ,v)\mapsto \lambda v\) such that

    • • \((\lambda +\mu )v=\lambda v+\mu v\), for all \(v\in V\), \(\lambda ,\mu \in \F \).

    • • \(\lambda (v+w)=\lambda v+\lambda w\), for all \(v,w\in V\), \(\lambda \in \F \).

    • • \((\lambda \mu )v=\lambda (\mu v)\), for all \(v\in V\), \(\lambda ,\mu \in \F \).

    • • \(1v=v\), for all \(v\in V\).

    We call the elements of \(\F \) scalars and those of \(V\) vectors.

  • Examples.

    • (1) Take \(V=\F \), the field itself, with addition and scalar multiplication the field addition and multiplication.

    • (2) \(\F ^n\), the \(n\)-fold Cartesian product of \(\F \) with itself, with component-wise addition and scalar multiplication:

      \begin{align*} \vec \lambda 1n+\vec \mu 1n&:=(\lambda _1+\mu _1,\dots ,\lambda _n+\mu _n)\\ \lambda \vec \lambda 1n&:=\vec {\lambda \lambda }1n. \end{align*}

    • (3) Let \(M_{m\times n}(\F )\) denote the set of \(m\) by \(n\) matrices (thus \(m\) rows and \(n\) columns) with entries in \(\F \). This is a vector space under entry-wise addition and scalar multiplication.

      Special cases are the vector spaces of column vectors \(M_{n\times 1}(\F )\) and row vectors \(M_{1\times n}(\F )\). In computations, we often identify \(\F ^n\) with \(M_{n\times 1}(\F )\) by associating \(x=\vec {x}1n\in \F ^{n}\) with the column vector

      \begin{equation*} \bx = \begin{pmatrix} x_1\\\vdots \\x_n \end {pmatrix}. \end{equation*}

    • (4) Here is a very general example: let \(\cI \) be any set and \(V\) a vector space. Recall that \(V^{\cI }\) denotes the set \(\set {f:\cI \to V}\) of all maps from \(\cI \) to \(V\).

      I claim that \(V^{\cI }\) is a vector space under pointwise addition and scalar multiplication. That is, for \(f,g:\cI \to V\) and \(\lambda \in \F \), we define

      \begin{align*} (f+g)(i)&:=f(i)+g(i)\\ (\lambda f)(i)&:=\lambda (f(i)), \end{align*} for all \(i\in \cI \).

      The zero element is just the constant zero function:

      \begin{equation*} 0(i):=0, \end{equation*}

      and the additive inverses are defined pointwise also:

      \begin{equation*} (-f)(i):=-(f(i)). \end{equation*}

      • Exercise.1 Prove the claim! That is, show that \(V^{\cI }\) is a vector space under pointwise addition and scalar multiplication.

      1 Question 4 on sheet 1.

  • Remark. For suitable \(\cI \), this last example captures many familiar vector spaces. For example:

    • • We identify \(\F ^n\) with \(\F ^{\set {\rng 1n}}\) by associating \(\vec {x}1n\in \F ^{n}\) with the map \((i\mapsto x_i)\).

    • • Similarly, we identify \(M_{m\times n}(\F )\) with \(\F ^{\set {\rng 1m}\times \set {\rng 1n}}\) by associating the matrix \(A\) with the map \((i,j)\mapsto A_{ij}\).

    • • \(\R ^{\N }\) is the set of real sequences \(\set {(a_{n})_{n\in \N }: a_n\in \R }\) that played such a starring role in Analysis 1.