Question

Suppose T: ℝ³→ℝ² is a linear transformation. Three vectors U₁, U₂ and U₃ are given below together with their images by T. Find T(W) for the given vector W.

U₁ = [1 1] U₂ = [3 3] U₃ = [1 2 ] W = [0 2 ] T(U₁) = [6 6 7] T(U₂) = [18 18 21] T(U₃ = [10 10 12] T(W) = [_ _ _]

Answer 1

To find T(W), express W as a linear combination of U₁, U₂, and U₃ and substitute the values into the images of U₁, U₂, and U₃ under T. T(W) = [18 18 21].

Step-by-step explanation:

To find the image of vector W under the linear transformation T, we need to express W as a linear combination of U₁, U₂, and U₃. We can write W = aU₁ + bU₂ + cU₃, where a, b, and c are scalars.

To find the values of a, b, and c, we can solve the system of equations:

a = 0

b = 2

c = 0

Substituting these values into the images of U₁, U₂, and U₃ under T, we get:

T(W) = aT(U₁) + bT(U₂) + cT(U₃) = 0 + [18 18 21] + 0 = [18 18 21].