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Ex. 9.2. Give an orthonormal basis for the 2-dimensional complex subspace V

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Linear algebra 2: exercises for Section 9 (first part)

Ex. 9.1. Let V be the vector space of continuous complex-valued functions defined on the interval [0, 1], with the inner product hf, gi = R

1

0

f (x)g(x) dx. Show that the set {x 7→ e

2πikx

: k ∈ Z} ⊂ V is orthonormal. Is it a basis of V ?

Ex. 9.2. Give an orthonormal basis for the 2-dimensional complex subspace V

3

of C

3

given by the equation x

1

− ix

2

+ ix

3

= 0.

Ex. 9.3. For the real vector space V of polynomial functions [−1, 1] → R with inner product given by

hf, gi = Z

1

−1

f (x)g(x)dx, apply the Gram-Schmidt procedure to the elements 1, x, x

2

, x

3

.

Ex. 9.4. For the real vector space V of continuous functions [−π, π] → R with inner product given by

hf, gi = 1 π

Z

π

−π

f (x)g(x)dx show that the functions

1/ √

2, sin x, cos x, sin 2x, cos 2x, . . .

form an orthonormal set. [Note: for any function f the inner products with this list of functions is the sequence of Fourier coefficients of f .]

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