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Ex. 9.1. For which values of α ∈ C is the matrix α

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Linear algebra 2: exercises for Section 9

Ex. 9.1. For which values of α ∈ C is the matrix  α

12

1

2

α



unitary?

Ex. 9.2. 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. 9.3. 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. 9.4. Let A be an orthogonal n × n matrix with entries in R. Show that det A = ±1.

If A is be an orthogonal 2 × 2 matrix with entries in R and det A = 1, show that A is a rotation matrix

A =  cos θ − sin θ sin θ cos θ



for some θ ∈ R.

Ex. 9.5. 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. 9.6. For the real vector space V of continuous functions [−π, π] → R with inner product given by

hf, gi = 1 π

Z

1

−1

f (x)g(x)dx show that for any n ≥ 0 the functions

1/ √

2, sin x, cos x, sin 2x, cos 2x, . . . , sin nx, cos nx form an orthonormal set.

1

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