GATE 2022 MA – Question 56

Linear Algebra · Vector Spaces and Linear Transformations · 2 marks · Multiple select

In ℝ³ set W₁={x:x₁+2x₂−x₃=0},W₂={x:2x₁+3x₃=0}. Which hold?

dim W₁=dim W₂
dim W₁+dim W₂−dim ℝ³=1
dim(W₁+W₂)=2
dim(W₁∩W₂)=1

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Show answer and explanation

Correct answer: (A) dim W₁=dim W₂; (B) dim W₁+dim W₂−dim ℝ³=1; (D) dim(W₁∩W₂)=1

Explanation

Each is a plane, dimension 2. Their normal vectors (1,2,−1) and (2,0,3) are independent, so the common kernel has dimension 3−2=1. The dimension formula gives dim(W₁+W₂)=2+2−1=3. **A,B,D**.

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