Question

Show that the transformation

t−R ³ →R ² , where t(x,y,z)=(x,y,z), is linear.

A. The transformation is linear.
B. The transformation is not linear.
C. Linear transformation cannot be determined.
D. This is not a valid transformation.

Answer 1

The transformation t(x,y,z)=(x,y,z) is linear because it satisfies the properties of linearity.

Step-by-step explanation:

To show that the transformation t−R³ →R², where t(x,y,z)=(x,y,z), is linear, we need to demonstrate that it satisfies the properties of linearity. For a transformation to be linear, it must satisfy two conditions:

1. t(u + v) = t(u) + t(v), for all vectors u and v
2. t(cu) = ct(u), for all vectors u and scalar c

Let's use these conditions to prove the linearity of the given transformation:

1. t(u + v) = (u + v) = (u + v) = (u) + (v) = t(u) + t(v)
2. t(cu) = (cu) = c(u) = ct(u)

Since the transformation satisfies both conditions, we can conclude that the transformation t(x,y,z)=(x,y,z) is linear.