Matrix representation of a linear transformation

To find the matrix representation of a linear transformation, we need to consider the transformation of each basis vector and express the result in terms of the new basis vectors. In this case, we have a linear transformation T: R³ → R² defined by T(x, y, z) = (2x - 4y + 9z, 5x + 3y - 2z).

Step 1: Define the basis vectors

In the usual basis for R³, we have three basis vectors:
- e₁ = (1, 0, 0)
- e₂ = (0, 1, 0)
- e₃ = (0, 0, 1)

In the usual basis for R², we have two basis vectors:
- f₁ = (1, 0)
- f₂ = (0, 1)

Step 2: Determine the transformation of each basis vector

We apply the linear transformation T to each basis vector and express the result in terms of the new basis vectors.

- T(e₁) = T(1, 0, 0) = (2(1) - 4(0) + 9(0), 5(1) + 3(0) - 2(0)) = (2, 5)
- T(e₂) = T(0, 1, 0) = (2(0) - 4(1) + 9(0), 5(0) + 3(1) - 2(0)) = (-4, 3)
- T(e₃) = T(0, 0, 1) = (2(0) - 4(0) + 9(1), 5(0) + 3(0) - 2(1)) = (9, -2)

Step 3: Express the transformation in terms of the new basis vectors

The transformation of each basis vector can be expressed as a linear combination of the new basis vectors.

- T(e₁) = (2, 5) = 2f₁ + 5f₂
- T(e₂) = (-4, 3) = -4f₁ + 3f₂
- T(e₃) = (9, -2) = 9f₁ - 2f₂

Step 4: Construct the matrix representation

The matrix representation of the linear transformation T can be obtained by arranging the coefficients of the new basis vectors in a matrix.

|2 -4 9|
|5 3 -2|

The matrix representation of the linear transformation T: R³ → R², relative to the usual basis, is given by the 2x3 matrix shown above.