Part CCSIR NET December 2024matching-coefficients-gives-c3-equals-c2-either-way

Matching coefficients gives c3 equals c2 either way

Let (V, ⟨ , ⟩) be an inner product space over $C$ and T : V → V be $a C -$ linear transformation. Let $v_{1}$ and $v_{2}$ be non-zero vectors in V such that Tv $_{1} = c_{1} v_{1}$ and Tv $_{2} = c_{2} v_{2}$ for some $c_{1}, c_{2} \in C$ . Let $v_{3} = v_{2} - ($ ⟨ $v_{2}, v_{1}$ ⟩/‖ $v_{1}$ ‖ $^{2}) v_{1}$ . Suppose that $v_{3}$ is non-zero and Tv $_{3} = c_{3} v_{3}$ for some $c_{3} \in C$ . Which of the following statements are true?

A.The set { $v_{1}, v_{2}$ } is linearly independent.✓
B.If $c_{1} \neq = c_{2}$ , then ⟨ $v_{1}, v_{2}$ ⟩ = 0.✓
C.If $c_{1} = c_{2}$ , then $c_{3} = c_{2}$ .✓
D.If $c_{1} \neq = c_{2}$ , then $c_{3} = c_{2}$ .✓

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

See pricing

50 are analysed free — try those first.

Matching coefficients gives c3 equals c2 either way

More on this topic