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Orthogonality and Orthonormality

Jan 06, 2025, 6 min read

  • Let be an inner product space. A vector is a unit vector if . are orthogonal if .

    A subset is orthogonal if any two distinct elements of are orthogonal

    A subset is orthonormal if is orthogonal and consists entirely of unit vectors.

    • If then is orthonormal if and only if
      Where is the Kronecker delta.

  • (Friedberg e6.1.10) Pythagorean Theorem Analogue If are orthogonal elements of then

  • (Friedberg e6.1.12) Let be an orthogonal set in and . Then

  • (Friedberg 6.3) Let be an inner product space and let be an orthogonal set of nonzero vectors. If

    Then

    • (Friedberg 6.3.1) If above is orthonormal, then
    • (Friedberg 6.3.2) Let be an inner product space. Let be an orthogonal set of nonzero vectors. Then is linearly independent.

  • (Friedberg 6.4) Let be an inner product space and let be a linearly independent subset of . Define where and

    For

    Then is an orthogonal set of nonzero vectors such that .

    This construction is called the Gram-Schmidt Orthogonalization Process.

    • An orthonormal basis can then be obtained by applying the Gram-Schmidt Orthogonalization process and dividing by the norm of the vectors.

  • (Friedberg 6.5) Let be a finite dimensional inner product space. Then has an orthonormal basis . Furthermore, if and . Then

    • (Friedberg 6.5.1) Let be an orthonormal basis and a linear operator on and let . Then

  • Let be an orthonormal subset of an inner product space and . The Fourier coefficients of relative to are the scalars where .

  • Let be an inner product space and . Then is the orthogonal complement of defined as the set of all vectors in orthogonal to every vector in . That is

  • (Friedberg 6.6) Let be a finite dimensional subspace of an inner product space . If is an orthonormal basis of and then

    Where . This representation of is unique. That is is unique.

    • (Friedberg 6.6.1)

      Is the unique vector in that is “closest” to . That is, if then

      And also .

      We call the orthogonal projection of on

    • (Friedberg e6.2.6) If there exists such that but .

      • Intuition: Use (Friedberg 6.6). We have that assuming and by the fact that (otherwise ), we have

  • (Friedberg 6.7) Suppose that is an orthonormal set in an -dimensional inner product space . Then

    • can be extended to an orthonormal basis for using additional elements
    • If then the set is an orthonormal basis for
    • If , then . In fact (Fraleigh 6.2.12d) (see here).

  • (Friedberg e6.2.11) Let , where is an inner product space. Then

  • (Friedberg e6.2.12) Let be an inner product space, and be a finite dimensional subspace of . We have

    • so that
    • .

  • (Friedberg e6.2.13a) Parseval’s Identity. Let be an orthonormal basis for . For any we have

  • (Friedberg e6.2.14) Bessel’s Inequality Let be an inner product space and be any orthonormal subset of . For any we have

  • Let be an inner product space and be a projection. is an orthogonal projection if

  • Orthogonal Projections are uniquely determined by their range.

  • (Friedberg 6.23) Let be an inner product space and be a linear operator on . Then is an orthogonal projection if and only if

Geometric Interpretation

  • Let be a linear operator on a finite-dimensional real inner product space . is a rotation if is the identity on or if there exists a two dimensional subspace of , an orthonormal basis for and a real number such that

    And for all .

    We refer to as the axis of rotation. is called a rotation of about .

  • Let be a linear operator on a finite-dimensional real inner product space . The operator is called a reflection if there exists a one-dimensional subspace of such that

    We say that is a reflection of about .

  • (Friedberg 6.38) Let be an orthogonal operator on a two dimensional real inner product space . Then is either a reflection or a rotation.

    Also is a reflection if and only if the Determinant is a rotation if and only if .

    • (Friedberg e6.10.8) No orthogonal operator can be both a rotation and a reflection.

  • (Friedberg 6.38.1) Let be a two dimensional real inner product space. The composition of a reflection and a rotation on is a reflection on .

  • (Friedberg 6.39) Let be an orthogonal operator on a nonzero finite dimensional real inner product space . Then there exists a collection of pairwise orthogonal -invariant subspaces of such that

    • The restriction of to is either a rotation or a reflection.
    • (Friedberg 6.40)
      • The number of ’s for which is a reflection is even or odd according to whether or .
      • It is always possible to decompose as shown so that the number of ’s for which is a reflection is zero or one according to whether or . Also if is a reflection then .

  • (Friedberg 6.40.1) Let be an orthogonal operator on a finite dimensional real inner product space . Then there exists a collection of orthogonal operators such that

    • Each is either a rotation or a reflection
    • For at most one , is a reflection.

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