Question

Suppose that T:V→V is a linear transformation. 1. Prove that S(v)=T(T(v)) is also a linear transformation

Answer 1

To prove that S(v) = T(T(v)) is a linear transformation, we demonstrate that S satisfies the two main properties of linear transformations: it is linear with respect to both vector addition and scalar multiplication. This is confirmed by applying T's linearity properties twice.

Step-by-step explanation:

To prove that S(v) = T(T(v)) is a linear transformation, given that T:V→V is a linear transformation, we need to demonstrate two properties:

1. Linearity with respect to vector addition: S(u + v) = S(u) + S(v) for all vectors u, v in V.
2. Linearity with respect to scalar multiplication: S(c⋅v) = c⋅S(v) for all vectors v in V and all scalars c.

Since T is linear, T(u + v) = T(u) + T(v) and T(c⋅v) = c⋅T(v). Now, let's check these two properties for S:

1. S(u + v) = T(T(u + v)) = T(T(u) + T(v)) = T(T(u)) + T(T(v)) = S(u) + S(v)

   This verifies the linearity with respect to vector addition.

2. S(c⋅v) = T(T(c⋅v)) = T(c⋅T(v)) = c⋅T(T(v)) = c⋅S(v)

   This verifies the linearity with respect to scalar multiplication.

Therefore, S satisfies both properties of a linear transformation, proving that S(v) = T(T(v)) is indeed linear.