Let:T : ℝ3→ℝ3 be the transformation that projects each vector x=(x1,x2,x3)onto the plane x2=0,
so
T(x)=(x1 ,0 ,x3). Show that T is a linear transformation.
The first property for T to be linear is
T(0) =
Check if this property is satisfied for T. T(x1, x2, x3) =(x1, 0,x3)
T(0,0,0) = ( ,, _,) ,
So, is the first property satisfied?
Yes 0r No
The second property for T to be linear is T(cu+dv)=(see choices below)for all vectors(u and v)i n the domain of T and all scalars c, d
cT(u)+dT(v)
dT(u)+cT(v)dT(u)−cT(v)
cT(u)−dT(v)
for all vectors
u,
v in the domain of T and all scalars c, d.
Check if this property is satisfied for T. Let u=(u1, u2, u3) and v =(v1, v2, v3).
T(cu+dv)=(cu1+dv1, 0, cu3+dv3) =(cu1, , ) +(dv1, , )
Factor out the scalar in each ordered triple.
T(cu+dv) = (u1, 0, u3) + (v1, 0,v3)
Further simplify the previous equation.
T(cu+dv) = c
▼
(choices pick one)
T(v)
T(u)+d
▼
(choices for the arrow above pick one)
T(v)
T(u)
So, is the second property satisfied?
Yes
No
Thus, T ▼
is
is not
linear.