235_sample_mid_2_soln.pdf - Math 235 Sample Midterm 2 1...

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Math 235Sample Midterm 21.Short Answer Problemsa) SupposeL:VWis a linear mapping, whereVis a vector space with a basisB={~v1, . . . ,~vn}andWis a vector space with a basisC={~w1, . . . , ~wm}. DefineC[L]B, thematrix ofLwith respect to basesBandC.
b) LetB={~v1, . . . ,~vn}be orthonormal in an inner product spaceVand let~vVsuch that~v=a1~v1+· · ·+an~vn. Prove thatai=h~v,~vii.
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c) ConsiderP2with inner producthp(x), q(x)i=p(-1)q(-1) +p(0)q(0) +p(1)q(1). Findthe value ofh1-x-x2,1 +x2i.
d) Write a basis for the four fundamental subspaces ofA=
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2.Suppose the matrix of a linear mappingL:R3P2(R) with respect to the standardbases isA=1111010-11.Find the inverse ofAand use it to define the inverse linear mappingL-1:P2(R)R3.-12-11-101-11. This is the matrix ofL-1with respect to the standard basis. From here, we get

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