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Linear Algebra: Companion notes

Abhay Soman

Section 6.1 Definition of Quotient Space

Let \(V\) be a vector space over a field \(F\) and let \(W\leq V\) be its subspace. We denote the abelian group operation on \(V\) by ’\(+\)’ We consider the quotient group \(V/W\text{.}\) Recall that
\begin{equation*} V/W=\{v+W:v\in V\}, \end{equation*}
where
\begin{align*} v+W\amp=\{v+w:w\in W\}\\ \amp=\{u\in V:u-v\in W\}. \end{align*}
The set \(v+W\) is called the coset of \(v\) and \(v\) is called a coset representative for \(v+W\text{.}\) We denote the coset \(v+W\) by \([v]\text{.}\)

Definition 6.1.1. (Quotient space).

Let \(V\) be a vector space over a field \(F\) and \(W\leq V\) be a subspace. We denote the abelian group operation on \(V\) by ’\(+\)’. The quotient group \(V/W\) with following operations of addition and scalar multiplication is called the quotient space of \(V\) modulo \(W\).
  1. Addition: \((v_1+W)+(v_2+W)=(v_1+v_2)+W\) for any \(v_1,v_2\in V\)
  2. Scalar multiplication: \(\alpha(v+W)=(\alpha v)+W\) for any \(v\in V\) and any \(\alpha\in F\)
With this addition and the scalar multiplication \(V/W\) is a vector space over \(F\text{.}\)
  1. Check that with the addition and the scalar multiplication defined in Definition 6.1.1 the quotient space \(V/W\) is a vector space over \(F\text{.}\)
  2. Show that the additive identity of \(V/W\) is \(0+W\text{.}\)

Remark 6.1.3.

It is possible to realize the quotient space geometrically. We give a few examples.
  1. Consider \(V=\R^2\) and \(W=\{(a,a):a\in\R\}\text{,}\) a line passing through origin. Any coset \(v+W\) is a line passing through \(v\) and parallel to \(W\text{.}\)
  2. Consider \(V=\R^3\) and \(W=\{(a,b,0):a,b\in\R\}\text{,}\) a plane passing through origin. Any coset \(v+W\) is a plane passing through \(v\) and parallel to \(W\text{.}\)
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