GATE 2023 MA – Question 14

Linear Algebra · Vector Spaces and Linear Transformations · 1 mark · Multiple choice

A linear T:ℝ³→ℝ³ satisfies T(1,0,0)=(0,1,1),T(1,1,0)=(1,0,1),T(1,1,1)=(1,1,2). Which holds?

Injective but not onto
Injective and onto
Neither injective nor onto
Onto but not injective

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Show answer and explanation

Correct answer: (C) Neither injective nor onto

Explanation

Subtract successive equations to obtain columns (0,1,1),(1,−1,0),(0,1,1). The first and third coincide and the second is independent, giving rank 2 and nullity 1. **C**.

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