Inner Product Space

Exercise1-4

1. For vectors ${\bf A} = (-1,3,1)$ and ${\bf B} = (2,4,-3)$, find the followings: (a) $\Vert{\bf B}\Vert$ (b) ${\bf A}\cdot {\bf B}$ (c) Angle between A and B (d) Unit vector in the dierction of A

2. Determine which system is orthogonal. If it is orthogonal, find the orthonormal system.

(a) $\{(1,3),(6,-2)\}$

(b) $\{(1,2,2),(-2,2,-1),(2,1,-2)\}$

(c) $\{{\bf i} - 2{\bf j} + 3{\bf k}, 2{\bf i} - \frac{1}{2}{\bf j} - \frac{1}{3}{\bf k}, 3{\bf i} + 3{\bf j} + {\bf k}\}$

3. Find an equation of plane going thru a point $(5,-1,3)$ and normal vector is 2i + j - k.

4. Let A,B be space vectors. Then prove the following inequality:

$$\vert{\bf A}\cdot{\bf B}\vert \leq \Vert{\bf A}\Vert \Vert{\bf B}\Vert .$$

This result is called Cauchy-Schwarz inequality．

5. LetA, B, C be space vectors. Then show the following inequality:

$$\Vert{\bf A} - {\bf B}\Vert \leq \Vert{\bf A} - {\bf C}\Vert + \Vert{\bf C} - {\bf B}\Vert .$$

6. Let $f(x),g(x)$ be a function vector in $PC[a,b]$．Show the following:

$$\vert(f,g)\vert \leq \Vert f\Vert \Vert g\Vert .$$

This result is called Schwarz inequality．

7. For $PC[0,2]$, Find the norm of the followings:．

(a) $f(x) = x$ (b) $f(x) = \sin{\pi x}$ (c) $f(x) = \cos{\pi x}.$

8. Next three polynomials are called Legendre polynomial．

$$P_{0}(x) = 1, \ \ P_{1}(x) = x, \ \ P_{2}(x) = \frac{3x^2 - 1}{2}$$

Show that these polynomials are orthogonal system in $PC[-1,1]$.