Vectors over real n-space, Inner products, Orthogonal vectors and spaces, Triangle Inequality, Schwarz Inequality, Gram-Schmidt orthogonalization process, Gramian Matrix

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We will, on this page, present results for vectors over real n-space, V_n(R). Vector elements and scalars are real numbers from the field of real numbers, R.

Inner (or scalar) product of two real n-vectors. Let

and

be two vectors whose elements are real numbers. Then their inner product is given by

The inner product is called the dot product in vector analysis.

Note that the concept of the inner product of vectors of n-space represents a generalization of the concept of the dot product of vectors in three-dimensional space. Much in the theory of n-space is a generalization of concepts of ordinary three-dimensional space.

Properties of inner products. Let X, Y and Z be real n-vectors and c be a real number. Then the following laws hold:

1. X•Y = Y•X Commutative Law

2. (cX)•Y = c(X•Y)

3. X•(Y + Z) = X•Y + X•Z Distributive Law

Orthogonal vectors. Two vectors in n-space are said to be orthogonal if their inner product is zero.

The concept of orthogonality of vectors in n-dimensional space represents a generalization of the concept of perpendicularity of vectors in two and three-dimensional space. Vectors in ordinary two and three-dimensional space are perpendicular if and only if their inner products (or dot products) are zero.

Length of a real vector. The length of a real n-vector

is denoted by ||X|| and defined as

The concept of the length of vectors in n-space represents a generalization of the concept of length in two and three-dimensional space. In the case of two and three-dimension space it corresponds to the usual concept of length.

A vector X whose length is ||X|| = 1 is called a unit vector. The elementary vectors E_i are examples of unit vectors.

Theorem. For two real n-vectors X and Y:

Proof

The Triangle Inequality. For two real n-vectors X and Y

The Schwarz Inequality. For two real n-vectors X and Y

Proof

Orthogonal vectors and spaces. The concept of n mutually orthogonal vectors of

n-space corresponds to the concept of a set of three mutually perpendicular vectors in 3-space.

Theorems

1] Any set of m mutually orthogonal non-zero vectors of n-space is a linearly independent set and spans an m-dimensional subspace of n-space.

Example. Let n = 3 and m = 2 in the above theorem. It then says that any set of 2 mutually orthogonal non-zero vectors of 3-space is a linearly independent set and spans a 2-dimensional subspace of 3-space (i.e. a plane).

A vector is said to be orthogonal to a vector space if it is orthogonal to every vector of the space. (For an intuitive example of this consider a plane passing through the origin in 3-space and a vector that is perpendicular to it. The vector is orthogonal to the two-dimension subspace of 3-space represented by the plane – and orthogonal to every vector in it.).

2] If a vector is orthogonal to each of the vectors X₁, X₂, ... ,X_m of n-space it is orthogonal to the space spanned by them.

3] Let V_n^h(R) be a h-dimensional subspace of a k-dimensional subspace V_n^k(R) of real n-space V_n(R) where h < k. Then there exists at least one vector X of V_n^k(R) which is orthogonal to V_n^h(R) .

Example. Pass a plane through the origin of a Cartesian system in three dimensional space and label it K. Let L be a line in plane K that passes through the origin. Let V_n^h(R) be line L and V_n^k(R) be plane K and V_n(R) be 3-space. Then the theorem says that plane K must contain at least one line perpendicular to line L.

4] Every m-dimensional vector space contains exactly m mutually orthogonal vectors.

5] Two vector spaces are said to be orthogonal if every vector of one is orthogonal to every vector of the other space.

Example. A plane passing through the origin in 3-space and a line that is perpendicular to it.

6] The set of all vectors orthogonal to every vector of a given k-dimensional vector subspace V_n^k(R) of n-space is a unique (n - k)-dimensional subspace V_n^n-k(R) .

Unit vectors in n-space. Normalization of vectors. We may associate with any vector X ≠ 0 a unique unit vector U obtained by dividing the components of X by ||X|| . This operation is called normalization.

Example. To normalize the vector

divide each component by

and obtain the unit vector

Orthogonal bases. A basis of a vector space (or subspace) that consists of mutually orthogonal vectors is called an orthogonal basis of the space.

Orthonormal bases. If the mutually orthogonal vectors of an orthogonal basis of a space are also unit vectors the basis is called a normal orthogonal or orthonormal basis.

The Gram-Schmidt orthogonalization process. Suppose X₁, X₂, .... ,X_m constitute a basis of some vector space. The Gram-Schmidt orthogonalization process is a procedure for generating from these m vectors an orthogonal basis for the space. The process involves computing a sequence Y₁, Y₂, .... ,Y_m inductively as follows:

or, stated more succinctly,

Derivation of process

The vectors Y₁, Y₂, .... ,Y_m given by the above algorithm are mutually orthogonal but not orthonormal. To obtain an orthonormal sequence replace each Y_i by

The Gramian Matrix. Let X₁, X₂, .... ,X_p be a set of real n-vectors. The Gramian matrix is defined as

where X_i∙X_j is the inner product of X_i and X_j .

A set of vectors X₁, X₂, .... ,X_p are mutually orthogonal if and only if their Gramian matrix is diagonal.

For a set of real n-vectors X₁, X₂, .... ,X_p, the determinant of the Gramian matrix |G| has a value |G| 0. The set of vectors are linearly dependent if and only if |G| = 0.

Orthogonal matrices. An orthogonal matrix is defined as a square matrix that is equal to the inverse of its transpose. This stated condition

is equivalent to the condition

Thus we see that an orthogonal matrix is one whose inverse is equal to its transpose.

Theorems

1] The column vectors of an orthogonal matrix are mutually orthogonal unit vectors. The same is true for its row vectors – they also are mutually orthogonal unit vectors.

2] If the real n-square matrix A is orthogonal, its column vectors (or row vectors) are an orthonormal basis of V_n(R), and conversely.

3] The inverse and transpose of an orthogonal matrix are orthogonal.

4] The product of two or more orthogonal matrices is orthogonal.

5] The determinant of an orthogonal matrix has a value of +1 or -1.

Orthogonal transformations. A linear transformation Y = AX is called orthogonal if its matrix A is orthogonal.

Theorems

1] A linear transformation Y = AX preserves lengths if and only if it preserves inner products.

Let a linear transformation Y = AX carry n-vectors X₁ into X₂ into Y₁ and Y_2, respectively. Then the linear transformation preserves lengths if and only if X₁ • X₂ = Y₁ •Y₂ .

2] A linear transformation preserves lengths if and only if it is orthogonal.

In three dimensional space an orthogonal transformation corresponds to a rotation of the coordinate system. In n-dimensional space an orthogonal transformation corresponds to the n-dimensional equivalent of a coordinate system rotation.

3] If the linear transformation Y= AX is a transformation of coordinates from the E-basis to another, the Z-basis, then the Z-basis is orthogonal if and only if A is orthogonal.

References.

Ayres. Matrices (Schaum).