Question

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,

2. (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.

3. (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.

4. (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.

Answer 1

TIN IV L.T Such that T, Ty To &haw Ts one-one Cet XY Nsuch that n4 yy 0 TBn the def" of product Ди иеn Tr. Ty> Hence, TB Ona- to-one Now = (x, x+y)+ (9,x+ ) x,x , ,+414 the Ldefof Inuar product 20,4 -T910 (*iS) ne lovened from H2uLo, inner product can be the non eEF A Given that, Bv,V,- - Va Babasis. matnix ith columus v,,v2,-Vn Nef. A 02 the G AA A the trauspose of A Then, inven Hi ble To Show GB to prove ontradiction. we

If possible, let S uot inventi ble 141o ATA IATIIAEO 1A12 JA1 ,V3,.VnY 1neanly defend ent auich fact tuat F = v,- V are tontradicts a basis( Fs L.I) Hence, inventible oved) to the