All bases considered in these are assumed to be ordered bases. In Exercise, compute coordinate vector v with respect to the giving basis S for V.

Efan Halliday 2021-02-25 Answered
All bases considered in these are assumed to be ordered bases. In Exercise, compute the coordinate vector of v with respect to the giving basis S for V. VisP1,S=t+1,t2,v=t+4
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lobeflepnoumni
Answered 2021-02-26 Author has 99 answers
We are given the following ordered basis S for the vector spaceV=P1 as well as the following vector v in V:S=t+1,t2,v=t+4We have to compute the coordinate vector [v]S, of v with respect to the basis S.We havev=t+4
=a(t+1)+b(t2)
=(a+b)t+(a2b).Equating coefficients yields the following linear system:a+b=1
a2b=4The associated augmented matrix for this system isA=[111124]Subtracting the first row from the second yields[111033]Dividing the second row by -3 yields[111011]Finally, subtracting the second row from the first yields the following reduced row echelon form ofA:
AR=[102011]Thus, we have the solutiona=2
b=1Therefore, the coordinate vector [v]S of v respect to the basis S is [v]S=[21]
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