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- Homework Statement
- Please see below

- Relevant Equations
- Span

Linear Independent

I think I can prove W is a subspace of V. I want to ask about basis of W.

Let $$V = a_1+a_2 \sin t+a_3 \cos t +a_4 \sin (2t)+a_5 \cos (2t)$$

$$W = p(t) = q"(t) + q(t)$$

$$=-a_2 \sin t-a_3 \cos t-4a_4 \sin (2t)-4a_5 \cos(2t)+a_1+a_2 \sin t+a_3 \cos t +a_4 \sin (2t)+a_5 \cos (2t)$$

$$=a_1-3a_4 \sin (2t) -3a_5 \cos (2t)$$

Since all elements in W are linearly independent, the basis for W is {1, sin (2t), cos (2t)}

Am I correct? Thanks

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