Question

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

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

T(W) = [ _ ]
[ _ ]
[ _ ]

Answer 1

To find T(W), substitute the images of vectors U₁, U₂, and U₃ into the equation using the linearity property of T.

Step-by-step explanation:

To find T(W), we need to find the image of vector W under the linear transformation T. We are given the images of vectors U₁, U₂, and U₃ under T. To find T(W), substitute the images of vectors U₁, U₂, and U₃ into the equation using the linearity property of T.

First, we can write W as a linear combination of U₁, U₂, and U₃: W = 0U₁ + 3U₂ + 1U₃.

Using the linearity property of T, we can then find T(W) by substituting the images of U₁, U₂, and U₃ into the equation: T(W) = 0T(U₁) + 3T(U₂) + 1T(U₃) = 0[6] + 3[18] + 1[10] = [0 + 54 + 10] = [64].