Inner Product Spaces: Exercise

INNER PRODUCT SPACES: EXERCISE

1. In C ([0, 1]), let f(t)= t and g(t) = e^t, Compute <ƒ, g >, || ƒ ||, || g || and ||f + g||.

Then verify both the Cauchy Schwarz inequality and the triangle inequality.

[Ans: <f,g> = 1, || f || = √3/3, || g || = √[(e²-1)/2], || f + g || = √[(11+3e²)/6]

2. Use the Frobenius inner product to compute || A ||, ||B || and <A, B> for 

3. In C², show that <x, y>=x Ay* an inner product, where 

Compute <x, y> for x=(1−i, 2+3i) and y = (2+ i, 3−2i).

4. Apply the Gram−Schmidt process to the given subset S of the inner product space V to obtain an orthogonal basis for span(S). Then normalize the vectors in this basis to obtain an orthonormal basis B for span(S) and compute the Fourier coefficients of the given vector relative to B.

(a) V=R³, S={(1, 0, 1), (0, 1, 1), (1, 3, 3)} and x=(1, 1, 2)

(b) V=R³,S={(1, 1, 1), (0, 1, 1), (0, 0, 1)} and x=(1, 0, 1)

[Ans: {√3/3 (1, 1, 1), √6/6 (−2, 1, 1), √2/2 (0, ‒1, 1) } ; 2√3/2, ‒√6/6, √2/2

(c) V=P₂(R) with the inner product

 < f(x), g(x)> = ₀ʃ¹ f(t) g(t) dt, S= { 1, x, x²) and h(x)=1+x.

 [Ans: { 1,2√3(x – 1/2), 6√3 (x² – x + 1/6)}; 3/2.√3/6,0]

 (d) V=R⁴, S={(2, − 1, −2, 4), (−2, 1, −5, 5), (− 1, 3, 7, 11)} and x=(−11, 8, 4, 18)

 [Ans: { 1/5 (2,−1,−2,4) , 1/√30 (−4, 2,−3, 1), 1/√155 (−3, 4, 9, 7) }; 10, 3 √30, √155]

(f) V=R⁴, S = {(1, −2, −1, 3), (3, 6, 3, − 1), (1, 4, 2, 8)), and x = (− 1, 2, 1, 1).

5. For each of the following inner product spaces V (over F ) and linear transformations g: V→ F, find a vector y such that g(x) = ( x, y ) for all x∈ V.

(a) V=R³, g(a₁, a₂, a₃) = a₁ − 2a₂ + 4a₃

  [Ans: y = (1,−2, 4)]

(b) V=P₂(R) with <f, h> = ₀ʃ¹ f(t) h(t) dt, g(f) = f(0) + f '(1).

 [Ans: y=210x²−204x+33]

6. For each of the following inner product spaces V and linear operators T on V, evaluate T* at the given vector in V.

(a) V=R², T (a, b) = (2a + b, a−3b), x = (3, 5)   [Ans: T*(x) = (11, ‒12)]

(b) V=P₁(R) with <f, g> = _-1ʃ¹ f (t) g(t) dt, T(f) = ƒ' +3ƒ, ƒ (t) = 4 ‒ 2t

 [Ans: T*[f(t)]= 12+6t]

7. For each of the sets of data that follows, use the least squares approximation to find the beat fits with both (i) a linear function and (ii) a quadratic function. Compute the error E in both cases.

(a) {(1, 2), (3, 4), (5, 7), (7, 9), (9, 12)}

(b) {(−2,4), (−1, 3), (0, 1), (1, −1), (2,−3)}

[Ans: The linear function y = 1.25t +0.55 with E=0.3.

Quadratic function is t²/56 + 15t/14 + 239/280 with E=0.22857]