advertisement

Math 304 Quiz 14 Summer 2006 Linear Algebra Both problems concern the inner-product space C[0, 1] of continuous functions on the interval [0, 1] with the following inner product: Z 1 hf, gi = f (x)g(x) dx. 0 1. For which positive number a does the function f (x) = ax4 have norm equal to 1? Solution. Since kf k2 = hf, f i = 2 want a /9 = 1, so a = 3. R1 0 h 9 i1 a2 x8 dx = a2 x9 = a2 /9, we 0 2. Find the cosine of the angle between the functions f (x) = g(x) = 1. √ x and Solution. Since hf, gi = kf k kgk cos(θ), compute: 1 2 x3/2 = , x dx = hf, gi = 3/2 0 3 0 2 1 Z 1 1 1 x = , kf k2 = so kf k = √ , x dx = 2 0 2 2 0 Z 1 kgk2 = 12 dx = 1, so kgk = 1. Z 1 √ 0 Therefore √ hf, gi 2/3 2 2 cos(θ) = = √ = . kf k kgk 1/ 2 3 √ [The question did not ask for the value of θ, which is arccos(2 2/3). Your calculator will give you an approximate value of either 0.34 radians or 19.5 degrees.] June 22, 2006 Dr. Boas