INNER PRODUCT SPACE. Definition 1

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1 INNER PRODUCT SPACE Definition 1 Suppose u, v and w are all vectors in vector space V and c is any scalar. An inner product space on the vectors space V is a function that associates with each pair of vectors in V, say u and v, a real number denoted by that satisfies the following axioms: 1. [symmetry axiom] 2. [additivity axiom] 3. [homogeneity axiom] 4. A vector space with its inner product is called an inner product space. Euclidean inner product If and are vectors in R n, then the formula defines to be the Euclidean inner product on R n. The weighted Euclidean inner product If are positive real numbers, which we shall call weights, and if and are vectors in R n, then the formula defines an inner product on R n. It is called the weighted Euclidean inner product with weights Proof:. Suppose that u, v, and a are all vectors in R n and that c is a scalar. where are positive real numbers, ; and

2 First axiom we know that real numbers commute with multiplication we have, The second axiom The third axiom The last axiom because are positive real numbers so 0. And suppose that. So the fourth axiom is also satisfied. Therefore proved that defines an inner product on R n. It is called the weighted Euclidean inner product with weights ( are positive real number). Definition 2 Suppose that V is an inner product space. The norm or length of a vector u in V is defined to be,

3 Definition 3 Suppose that V is an inner product space and that u and v are two vectors in V. The distance between u and v, denoted by is defined to be, Example 1 Let f = f(x) and g = g(x) be two functions on C[a, b], the set of all continuous functions on [a, b]. Define Then defines an inner product on C[a, b]. Proof: Suppose that f, g, and h are continuous functions in C[a, b] and that c is any scalar. Here is the work showing the first axiom is satisfied. The second axiom Here s the third axiom Finally, the fourth axiom.

4 Now, recall that if you integrate a continuous function that is greater than or equal to zero then the integral must also be greater than or equal to zero. Hence, Next, if then clearly we ll have. Likewise, if we have then we must also have. Example 2 Let A be an invertible matrix. The following defines an inner product on R n Example 3 For A and B in R n Let then defines an inner product in R n. Proof: Suppose that are two matrices in. And an inner product space of we can express This formula is very similar to the Euclidean inner product formula. Therefore it s proved that A and B in R n an inner product of matrices defines the formula Cauchy-Schwarz inequality Suppose that any vectors u and v in a inner product space V then or

5 Property of Length (norm) If u and v are vectors in an inner product space V, and if k is any scalar, then a. b. c. d. (Triangle inequality) e. The angle θ between u and v is defined by Triangle Inequality If V is an inner product space with norm then. Proof: We have because hence So the Cauchy-Schwarz inequality implies that Therefore prove that

6 Property of distance If u, v, and w are vectors in an inner product space V, and if k is any scalar, then a. b. c. d. (Triangle inequality) Definition Orthogonality Suppose that u and v are two vectors in an inner product space then u and v are called orthogonal if Orthogonal sets and bases Suppose that V an inner product space and let V. W is called orthogonal set if any 2 member mutually orthogonal or for every i j with i,j=1,2,3,.k. And W is called orthonormal set if orthogonal set W and distance of any member of W is one or we can express. Phythagoras Theorem Let V be an inner product space. If are orthogonal then.

7 Proof: Since hence the v and w are orthogonal so. Therefore prove that the phytagoras theorem. Projection theorem If W is a finite-dimensional subspace of an inner product space V, then any vector u in V can be expressed in exactly one way as where is in W and is in W Orthogonal Projection Suppose that W is spanned by orthogonal set and orthogonal projection v to W is The reason in W and orthonormal set it s means the distance or norm the member of set is one or. And orthogonal v component to W is or

8 Gram-Schmidt process Let W be a subspace of an inner product space V. Suppose that be a basis of W, not necessarily orthogonal. An orthogonal basis we can construct from B as follows: or we can write The method of constructing the orthogonal vector is known as the Gram-Schmidt process.

9 Orthogonal matrix Let V be an n-dimensional inner product space. A linear transformation T : V V is called an isometry if for any, Theorem A linear transformation T : V V is an isometry if only if T preserving inner product, e.g for, Proof: for vectors identity with So it s clear that the length preserving is equivalent to the inner product preserving. An matrix Q is called orthogonal if i.e; Theorem Let Q be an matrix. The following are equivalent a) Q is orthogonal b) is orthogonal c) The column vectors of Q are orthonormal d) The row vectors of Q are orthonormal Diagonalizing real symmetric matrices Let V be an n-dimensional real inner product space. A linear mapping T : V V is said to be symmetric if for all

10 Let A be an real symmetric matrix. Let T: R n R n be defined by T(x)=A(x). Then T is symmetric for the Euclidean n-space. In fact we have An real symmetric matrix A is called positive definite if for any nonzero vector ; Complex inner product space Definition Suppose that V be an complex inner product space. An inner product space of V is a function : satisfying the following properties: 1. Linearity 2. Conjugate symmetric property 3. Positive definite property For any, For any, since is a nonnegative real number, the length of u to be real number Cauchy-Schwarz inequality Let V be a complex inner product space. Suppose that any vectors u and v in a inner product space V then or Like real inner product space, one can similarly define orthogonality, the angle between 2 vectors, orthogonal set, orthonormal set, orthogonal projection and Gram-Schmidt process, etc.

11 Complex Euclidean space C n For vectors and where defines Let be orthogonal basis of an inner product space V. Then for any ; Let W be a subspace of V with an orthogonal basis. Then orthogonal projection is given by

12 RESUME OF INNER PRODUCT SPACE Linear Algebra 2 by Rini Kurniasih K Mathematics Education 2010 Training Teacher and Education Faculty SEBELAS MARET UNIVERSITY (UNS) 2011

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