Let T be a unitary operater on an Euclidean space V. Show that (T(a)|T(B))=(a), for all a, BEV.

We can define a one-to-one correspondence between the elements of Pn and Rn by p(x) = a1 + a2x + a3x2 + … + anxn-1   (a1, …, an)T = a Show that if p  a and q  b, then (a) αp  αa for any scalarα. (b) p + q  a + b [In general, two vector spaces are said to be isomorphic if their elements can be put into a one-to-one correspondence that is preserved under scalar multiplication and addition as in (a) and (b).]

Consider the set R²={(a,b)Da,b ER}. Suppose the following operations are defined on this set. Addition: (X 1.y 1)® (x 2Y2)=(x1+X2.Y1+Y2) • Scalar Multiplication: C © (x.y)=(cx,÷y) Which of the following axioms for a vector space fail to hold under these operations? O Distributivity of scalar multiplication over vector addition (c +d)O u=(c ©u)e (dO u) for all scalars C,d and all ordered pairs u=(x.y) 10 u=u for all u ER? O Closure under addition Additive identity QUESTION 15 Find the coordinates of X= (13, 26, 39) relative to the orthonormal basis B = inR 13 49 29 -78 O b.(x], =| 58 39 61 =| 58

Suppose that the set A={X₁, X2..., Xn}, X₁ ER" spans the entire space R", i.e. for any arbitrary n -vector y E R",3 a set of coefficients {t₁, t₂,... tn} with at least one t; #0 such that 1tixi = y. Prove that the coefficients t₁, t₂,.., t are unique.

= 4. Let P₂ be the vector space of real polynomials of degree at most 2, that is P₂ = {ao + a₁ + a₂x²2 a, ER}. Consider the inner product on P₂ defined by Define T: P2 P₂ by (p\q) = p(x)q(x) dx. 0 T(ao + ax + a2x²) = a₁x. (a) Show that T is not Hermitian (i.e. self-adjoint). (b) Consider the basis B = {1, x, x2} for P₂. Show that the matrix [T]g is Hermitian. Explain why this does not contradict with result from Part (a).