Editing Quotient space (linear algebra) (section)

===Lines in Cartesian Plane===

Let {{nowrap|1=''X'' = '''R'''<sup>2</sup>}} be the standard [[Cartesian plane]], and let ''Y'' be a [[line (geometry)|line]] through the origin in ''X''.  Then the quotient space ''X''/''Y'' can be identified with the space of all lines in ''X'' which are parallel to ''Y''.  That is to say that, the elements of the set ''X''/''Y'' are lines in ''X'' parallel to ''Y''.  Note that the points along any one such line will satisfy the equivalence relation because their difference vectors belong to ''Y''. This gives a way to visualize quotient spaces geometrically. (By re-parameterising these lines, the quotient space can more conventionally be represented as the space of all points along a line through the origin that is not parallel to ''Y''. Similarly, the quotient space for '''R'''<sup>3</sup> by a line through the origin can again be represented as the set of all co-parallel lines, or alternatively be represented as the vector space consisting of a [[plane (geometry)|plane]] which only intersects the line at the origin.)