Let V be a finite dimensional vector space over R with an inner product 〈x, y〉 ∈ R for x, y ∈ V .
(a) (3points) Let λ∈R with λ>0. Show that
〈x,y〉′ = λ〈x,y〉, for x,y ∈ V,
(b) (2 points) Let T : V → V be a linear operator, such that
〈T(x),T(y)〉 = 〈x,y〉, for all x,y ∈ V.
Show that T is one-to-one.
(c) (2 points) Recall that the norm of a vector x ∈ V is defined by ||x|| = 〈x, x〉. Show that
〈x,y〉=1(||x+y||2−||x||2−||y||2), forallx,y∈V.2
Hence, the inner product can be recovered from the norm.
Hint: Rewrite 〈x + y, x + y〉 using the properties of inner
products. Use that 〈x, y〉 ∈ R is a real number by assumption.
(d) (3 points) Let β = {v1,...,vn} be a basis of V. The Gram matrix G ∈ Mn,n(R) of the inner product 〈−, −〉 with respect to β is defined by
Gi,j =〈vi,vj〉.
Show that G is invertible.
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