Question

Let V and W be subspaces, and let T:V→W be a linear transformation. If a is a scalar, define aT:V→W by [aT](v)=a[T(v)] for each v in V. Show that aT is a linear transformation.

Answer 1

To show that aT is a linear transformation, we need to prove that it satisfies the properties of linearity: additivity and homogeneity.

Step-by-step explanation:

To show that aT is a linear transformation, we need to prove that it satisfies the properties of linearity: additivity and homogeneity.

Additivity:

Let v₁, v₂ be vectors in V. We have [aT](v₁ + v₂) = a[T(v₁ + v₂)] = a[T(v₁) + T(v₂)] (since T is linear) = aT(v₁) + aT(v₂), which shows that aT is additive.

Homogeneity:

Let v be a vector in V and c be a scalar. We have [aT](cv) = a[T(cv)] = a[cT(v)] (since T is linear) = (ac)T(v), which shows that aT is homogeneous.

Since aT satisfies both additivity and homogeneity, it is a linear transformation.