Question

Let V and W be vector spaces, and let v1​,v2 ​∈ V and w1​,w2​ ∈ W.

a. Suppose T:V→W is a linear transformation. Find T(3v1​+v2​) and write your answer in terms of T(v1​) and T(v2​). (Enter T(v1​) as T(v1) and T(v2​) as T(v2).)

Answer 1

To find T(3v1 + v2), apply the properties of linear transformations: T(3v1 + v2) becomes 3T(v1) + T(v2), demonstrating homogeneity and additivity.

Step-by-step explanation:

To calculate T(3v1 + v2) for a linear transformation T:V→W, we apply the properties of linear transformations. These properties state that a linear transformation is both additive and homogeneous, meaning T(av+bu) = aT(v)+bT(u) where v, u are vectors and a, b are scalars. Applying these properties to our case:

Step 1: Apply the scalar multiplication property (homogeneity) - this gives us T(3v1) equals 3T(v1).

Step 2: Apply the vector addition property (additivity) - this asserts that T(v1 + v2) = T(v1) + T(v2).

Step 3: Combine the results of Step 1 and Step 2 using additivity - hence T(3v1 + v2) = 3T(v1) + T(v2).

This final result expresses T(3v1 + v2) in terms of T(v1) and T(v2).