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Lecture 25
Andrei Antonenko
April 7, 2003
1 Euclidean spaces
Definition 1.1. Let V be a vector space. Suppose to any 2 vectors v, u ∈ V there assigned a number from R which will be denoted by 〈v, u〉 such that the following 3 properties hold:
Bilinearity • 〈au 1 + bu 2 , v〉 = a〈u 1 , v〉 + b〈u 2 , v〉
- 〈u, av 1 + bv 2 〉 = a〈u, v 1 〉 + b〈u, v 2 〉
Reflexivity 〈u, v〉 = 〈v, u〉
Positivity 〈u, u〉 ≥ 0 ; moreover if 〈u, u〉 = 0, then u = 0.
Any function which satisfy properties above is called a scalar (inner) product. A vector space V with a scalar product is called a (real) Euclidean space.
Now we will give popular examples of the scalar products in different spaces.
Rn^ The scalar product of 2 vectors x and y from Rn^ can be defined as following: if
x = (x 1 , x 2 ,... , xn) and y = (y 1 , y 2 ,... , yn)
then 〈x, y〉 = x 1 y 2 + x 2 y 2 + · · · + xnyn. If the vectors are represented by column-vectors, i.e. by n × 1-matrices, then
〈x, y〉 = x>y.
Example 1.2. Let x = (1, 2 , 3) and y = (3, − 1 , 4). Then 〈x, y〉 = 1·3+2·(−1)+3·4 = 13.
Actually, there are other ways of defining a scalar product in the vector space Rn. For example, given positive numbers a 1 , a 2 ,... , an we can define the scalar product to be
〈(x 1 , x 2 ,... , xn), (y 1 , y 2 ,... , yn)〉 = a 1 (x 1 y 1 ) + a 2 (x 2 y 2 ) + · · · + an(xnyn).
Mm,n The scalar product of 2 m × n-matrices A and B such that
A =
a 11 a 12... a 1 n a 21 a 22... a 2 n
................... am 1 am 2... amn
and^ B^ =
b 11 b 12... b 1 n b 21 b 22... b 2 n
................... bm 1 bm 2... bmn
is equal to the sum of products of the corresponding entries: 〈A, B〉 = a 11 b 11 + a 12 b 12 + · · · + a 1 nb 1 n
- a 21 b 21 + a 22 b 22 + · · · + a 2 nb 2 n +...
- am 1 bm 1 + am 2 bm 2 + · · · + amnbmn. We can write this scalar product in terms of matrix multiplication, but to do it we need one more definition. Definition 1.3. The trace of the square matrix A is the sum of its diagonal elements. It is denoted by tr A. Example 1.4.
tr
Now, using trace we can write that 〈A, B〉 = tr(AB>). This is true, since diagonal elements of AB>^ are the following: (1, 1) : a 11 b 11 + a 12 b 12 + · · · + a 1 nb 1 n; (2, 2) : a 21 b 21 + a 22 b 22 + · · · + a 2 nb 2 n;
... (m, m) : am 1 bm 1 + am 2 bm 2 + · · · + amnbmn.
Pn(t), P (t), C[0, 1] Here we’re considering the spaces of polynomials and the space of con- tinuous functions on the interval [0, 1]. If f, g are 2 functions (or polynomials) we can define their scalar product by the following formula:
〈f, g〉 =
0
f (t)g(t) dt.
Moreover, we can consider the space of all functions which are continuous on the interval [a, b], and in this case the scalar product will be defined as
〈f, g〉 =
∫ (^) b
a
f (t)g(t) dt.
Positivity ‖v‖ ≥ 0. Moreover, ‖v‖ = 0 if and only if v = 0.
Linearity ‖kv‖ = |k|‖v‖.
Triangle inequality ‖u + v‖ ≤ ‖u‖ + ‖v‖.
Proof. Positivity: Directly follows from the definition of the norm. Linearity: ‖kv‖ =
〈kv, kv〉 =
k^2 〈v, v〉 = |k|
〈v, v〉 = |k|‖v‖. Triangle inequality: Let’s consider ‖u + v‖ ≥ 0. We can rewrite it in the following way:
‖u + v‖^2 = 〈u + v, u + v〉 = 〈u, u〉 + 2〈u, v〉 + 〈v, v〉 = ‖u‖^2 + 2〈u, v〉 + ‖v‖^2 by C-B-S inequality ≤ ‖u‖^2 + 2‖u‖‖v‖ + ‖v‖^2 = (‖u‖ + ‖v‖)^2
Now, taking roots of both sides we get
‖u + v‖ ≤ ‖u‖ + ‖v‖.
Geometrically speaking, the last inequality means that the side of the triangle is less then or equal to the sum of other 2 sides:
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