Let T be the linear operator on R3 defined by

T(xଵ, xଶ, xଷ) = (3xଵ + xଷ, −2xଵ + xଶ, −xଵ + 2xଶ + 4xଷ).

What is the matrix of T in the ordered basis {αଵ, αଶ, αଷ} where

αଵ = (1,0,1), αଶ = (−1,2,1), αଵ = (2,1,1)?

To find the matrix representation of the linear operator \( T \) in the given ordered basis \( \{ \alpha_1, \alpha_2, \alpha_3 \} \), we follow these steps:

### Step 1: Compute \( T(\alpha_1) \), \( T(\alpha_2) \), and \( T(\alpha_3) \)
We substitute each basis vector \( \alpha_i \) into \( T \) and calculate the result in terms of the standard basis \( \{e_1, e_2, e_3\} \), where:
\[
e_1 = (1, 0, 0), \quad e_2 = (0, 1, 0), \quad e_3 = (0, 0, 1).
\]

#### Compute \( T(\alpha_1) \):
\[
\alpha_1 = (1, 0, 1), \quad T(x_1, x_2, x_3) = (3x_1 + x_3, -2x_1 + x_2, -x_1 + 2x_2 + 4x_3).
\]
\[
T(\alpha_1) = T(1, 0, 1) = (3(1) + 1, -2(1) + 0, -1(1) + 2(0) + 4(1)) = (4, -2, 3).
\]

#### Compute \( T(\alpha_2) \):
\[
\alpha_2 = (-1, 2, 1), \quad T(x_1, x_2, x_3) = (3x_1 + x_3, -2x_1 + x_2, -x_1 + 2x_2 + 4x_3).
\]
\[
T(\alpha_2) = T(-1, 2, 1) = (3(-1) + 1, -2(-1) + 2, -(-1) + 2(2) + 4(1)) = (-2, 4, 9).
\]

#### Compute \( T(\alpha_3) \):
\[
\alpha_3 = (2, 1, 1), \quad T(x_1, x_2, x_3) = (3x_1 + x_3, -2x_1 + x_2, -x_1 + 2x_2 + 4x_3).
\]
\[
T(\alpha_3) = T(2, 1, 1) = (3(2) + 1, -2(2) + 1, -(2) + 2(1) + 4(1)) = (7, -3, 4).
\]

### Step 2: Express \( T(\alpha_1) \), \( T(\alpha_2) \), \( T(\alpha_3) \) in the basis \( \{ \alpha_1, \alpha_2, \alpha_3 \} \)
We solve for scalars \( c_1, c_2, c_3 \) such that:
\[
T(\alpha_i) = c_1 \alpha_1 + c_2 \alpha_2 + c_3 \alpha_3 \quad \text{for } i = 1, 2, 3.
\]
This involves solving a system of linear equations for each \( T(\alpha_i) \).

#### Solve for \( T(\alpha_1) = (4, -2, 3) \) in terms of \( \{ \alpha_1, \alpha_2, \alpha_3 \} \):
Let:
\[
4 \cdot e_1 - 2 \cdot e_2 + 3 \cdot e_3 = c_1(1, 0, 1) + c_2(-1, 2, 1) + c_3(2, 1, 1).
\]
This yields the system:
\[
c_1 - c_2 + 2c_3 = 4, \quad 2c_2 + c_3 = -2, \quad c_1 + c_2 + c_3 = 3.
\]

Solve this system to find \( c_1, c_2, c_3 \). Repeat for \( T(\alpha_2) \) and \( T(\alpha_3) \).

Would you like me to solve these systems explicitly?

Let T be the linear operator on R3 defined by T(x₁, x₂, x₃) = (3x₁ + x₃, −2x₁ + x₂, −x₁ + 2x₂ + 4x₃). What is the matrix of T in the ordered basis {α₁, α₂, α₃} where α₁ = (1,0,1), α₂ = (−1,2,1), α₃ = (2,1,1)?

Find the matrix representation of the linear operator T on R3 in the ordered basis {α₁, α₂, α₃}, where α₁ = (1, 0, 1), α₂ = (-1, 2, 1), and α₃ = (2, 1, 1). This solution covers the method of calculating T(α₁), T(α₂), and T(α₃), and solving for the coefficients in terms of the new basis.

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