Over the real field, let V=\(a,b,c) C3:a+ b=0,\ b+ c=0\. Which is a basis?

GATE 2020 ST – Question 12

Matrix Theory · Vector Spaces and Linear Systems · 1 mark · Multiple choice

Over the real field, let $V=\{(a,b,c)\in\mathbb C^3:a+\bar b=0,\ b+\bar c=0\}$. Which is a basis?

{(1,−1,1),(i,i,i)}
{(1,−1,1),(i,−i,i)}
{(1,−i,1),(i,1,i)}
{(1,−i,1),(i,1,−i)}

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Show answer and explanation

Correct answer: (A) {(1,−1,1),(i,i,i)}

Explanation

Let a=u+iv. The constraints give b=−u+iv and c=u+iv. Hence (a,b,c)=u(1,−1,1)+v(i,i,i), with u,v real. These two vectors satisfy the constraints and are real-linearly independent. Thus A.

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