Question

Match the LIST-I with LIST-II - For a linear transformation $T : U(F) \to V(F)$

Choose the correct answer from the options given below:

• A. A-II, B-III, C-IV, D-I
• B. A-III, B-I, C-IV, D-II
• C. A-III, B-II, C-I, D-IV
• D. A-III, B-II, C-IV, D-I

Hint:

Linear transformations always map zero vectors to zero vectors ($T(\mathbf{0}_U) = \mathbf{0}_V$) and preserve vector subtraction.

Answer 1

The Correct Option is D

Step 1 : Concept:
This question tests basic operational properties of linear transformations between vector spaces.

Step 2 : Key Formulas and Approach:
A mapping $T: U \to V$ is linear if:
1. $T(\alpha + \beta) = T(\alpha) + T(\beta)$ for all $\alpha, \beta \in U$.
2. $T(c\alpha) = c T(\alpha)$ for all $c \in F, \alpha \in U$.

Step 3 : Step-by-step Explanation:

• Item A:
$T(0) = T(0 + 0) = T(0) + T(0) \implies T(0) = 0$. Matches with III.

• Item B:
By definition of additivity, $T(\alpha + \beta) = T(\alpha) + T(\beta)$. Matches with II.

• Item C:
Using scalar multiplication property with $c = -1$: $T(-\alpha) = T((-1)\alpha) = -T(\alpha)$. Matches with IV.

• Item D:
Combining additivity and homogeneity: \[ T(\alpha - \beta) = T(\alpha + (-\beta)) = T(\alpha) + T(-\beta) = T(\alpha) - T(\beta) \] Matches with I.

Step 4 : Final Answer:
The correct matching is A-III, B-II, C-IV, D-I, which corresponds to option (D).