Question

Let \( T: \mathbb{R}^2 \rightarrow \mathbb{R}^2 \) be a linear transformation such that \( T\left( \begin{bmatrix} 1 \\ 2 \end{bmatrix} \right) = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \) and \( T\left( \begin{bmatrix} 3 \\ 2 \end{bmatrix} \right) = \begin{bmatrix} a \\ b \end{bmatrix} \). Then \( \alpha + \beta + a + b \) equals

• A. \( \frac{2}{3} \)
• B. \( \frac{4}{3} \)
• C. \( \frac{5}{3} \)
• D. \( \frac{7}{3} \)

Hint:

In linear transformations, express the transformation as a matrix equation to find the unknowns.

Answer 1

The Correct Option is A

Step 1: Understanding the linear transformation. 
The matrix representation of \( T \) must be found by solving the system of linear equations from the given information. Using the properties of linear transformations, you can derive the values for \( \alpha \), \( \beta \), \( a \), and \( b \). 
Step 2: Analyzing the options. 
\[ \alpha + \beta + a + b = \frac{2}{3} \] Step 3: Conclusion. 
The correct answer is (A) \( \frac{2}{3} \).