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Chapter 1 Linear algebra: key concepts

Let us warm up by revising some of the key ideas from Algebra 1B.

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

    \(V\times V\to V: (v,w)\mapsto v+w\) such that:

    • • \(v+w=w+v\), 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 \(v+(-v)=0=(-v)+v\).

    In fancy language, \(V\) with addition is an abelian group.

    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*}