Question

Let \( T : \mathbb{R}^2 \to \mathbb{R}^2 \) be a linear transformation defined by \[ T((1, 2)) = (1, 0) \quad \text{and} \quad T((2, 1)) = (1, 1). \] For \( p, q \in \mathbb{R} \), let \( T^{-1}((p, q)) = (x, y) \). Which of the following statements is TRUE?

• A. \( x = p - q; \quad y = 2p - q \)
• B. \( x = p + q; \quad y = 2p - q \)
• C. \( x = p + q; \quad y = 2p + q \)
• D. \( x = p - q; \quad y = 2p + q \)

Hint:

When working with linear transformations, express the output as a linear combination of known vectors, and solve for the coefficients to find the inverse.

Answer 1

The Correct Option is B

We are given that \( T((1, 2)) = (1, 0) \) and \( T((2, 1)) = (1, 1) \). To find the inverse of \( T \), we first express the vector \( (p, q) \) as a linear combination of the vectors \( (1, 2) \) and \( (2, 1) \). Let: \[ \begin{pmatrix} p \\ q \end{pmatrix} = a \begin{pmatrix} 1 \\ 2 \end{pmatrix} + b \begin{pmatrix} 2 \\ 1 \end{pmatrix}. \] This gives the system of equations: \[ p = a + 2b \quad \text{and} \quad q = 2a + b. \] Solving for \( a \) and \( b \) in terms of \( p \) and \( q \), we obtain: \[ a = p + q \quad \text{and} \quad b = 2p - q. \] Thus, \( x = p + q \) and \( y = 2p - q \), which corresponds to option (B). Thus, the correct answer is (B).