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

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:

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

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

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 X_{1}, X_{2}, ... ,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 X_{1}, X_{2}, .... ,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 Y_{1}, Y_{2}, .... ,Y_{m } inductively as follows:

or, stated more succinctly,

The vectors Y_{1}, Y_{2}, .... ,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 X_{1}, X_{2}, .... ,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 X_{1}, X_{2}, .... ,X_{p } are mutually orthogonal if and only if their Gramian matrix is
diagonal.

For a set of real n-vectors X_{1}, X_{2}, .... ,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

or

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 X_{1} into X_{2} into Y_{1} and Y_{2,} respectively. Then
the linear transformation preserves lengths if and only if X_{1} • X_{2} = Y_{1} •Y_{2} .

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).

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