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Problems #1,5 are graded.
Problem 1: Let (R4×1,⟨→x,→y⟩) be an inner product space where ⟨→x,→y⟩ denotes dot product. Show that the vectors →a=[1234] and →b=[−4−321] are orthogonal vectors.
Solution: Compute
⟨[1234],[−4−321]⟩=(1)(−4)+(2)(−3)+(3)(2)+(4)(1)=−4−6+6+4=0,
showing the vectors are orthogonal.
Problem 5: Consider the vector space (P,⟨⋅,⋅⟩) where the inner product is given by
⟨p(x),q(x)⟩=∫∞−∞p(x)q(x)e−x2dx.
It can be shown (via methods of problem 4) that the moments in this inner product space are
⟨1,1⟩=√π,
⟨x,1⟩=0,
⟨x2,1⟩=√π2,
⟨x3,1⟩=0,
⟨x4,1⟩=3√π4,
⟨x5,1⟩=0,
⟨x6,1⟩=15√π8.
Use these moments and the "linear in the first argument" property of inner products (noted here) to compute ⟨4x2+3x+9,1⟩ and ⟨32x5−64x3+24x,1⟩.
Solution: Recall that the "linear in the first argument" says that
⟨α→v+β→u,→w⟩=α⟨→v,→w⟩+β⟨→u,→w⟩.
Hence we may compute
⟨4x2+3x+9,1⟩=4⟨x2,1⟩+3⟨x,1⟩+9⟨1,1⟩=4√π2+0+9√π=11√π
and
⟨32x5−64x3+24x,1⟩=32⟨x5,1⟩−64⟨x3,1⟩+24⟨x,1⟩=0+0+0=0.
