In this figure, [math]M[/math] is a given transformation, [math]u[/math] is an arbitrary vector, [math]v[/math] is on a plane perpendicular to [math]u[/math], and the endpoint of [math]w[/math] is on a line parallel to [math]u[/math] so that [math]w=v+ku[/math] for some [math]k[/math]. Then [math]u' = Mu[/math], [math]v' = Mv[/math], and [math]w' = Mw=Mv+kMu[/math].[br][size=85][color=#0000ff]Note on moving an arbitrary point in 3D. Click on the endpoint of [math]u[/math], and arrows appear. Click again and see a different set of arrows. Four arrows show movement parallel to the [math]xy[/math]-plane. Two arrows show movement parallel to the [math]z[/math]axis. [br][/color][/size]
How does [math]w'[/math] behave "better" when [math]u[/math] is in the null space of [math]M[/math]?
When [math]u[/math] is in null space of [math]M[/math], [math]Mu=0[/math], which implies that [math]w'=Mv[/math], meaning that [math]w'[/math] depends only on [math]v[/math]. Geometrically, as [math]w[/math] moves along the dotted line on the left, [math]w'[/math] stays fixed on the right. [br][br]When [math]u[/math] is in [b]not[/b] in the null space of [math]M[/math], as [math]w[/math] moves along the dotted line on the left, [math]w'[/math] moves along a line parallel to [math]u'[/math].