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In the vector space for all real-valued functions, find a basis for the subspace spanned by {sint,sin2t,sintcost}.

Short Answer

Expert verified

{sinโกt,sinโกtcosโกt}and {sinโกt,sinโก2t}

Step by step solution

01

Find the set of vectors for V

Let v1=sinโกt, v2=sinโก2t, v3=sinโกtcosโกt.

Simplify the equation v3=sinโกtcosโกt using trigonometric identities.

v3=12(2sinโกtcosโกt)=12sinโก2t=12v2

So, the vectors v2 and v3 are dependent.

02

Write the spanning set

The spanning set reduces as shown below:

V=span{v1,v2,v3}=span{v1,v2}or=span{v1,v3}

So, the above set represents the basis of H.

So, the basis are {sinโกt,sinโกtcosโกt} and {sinโกt,sinโก2t}.

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