Math 220 - April 12, 1999

Inner Products and Orthonormal Bases

• Definition. An inner product on a vector space V is a real-valued function I of two variables from V that satisfies the following rules:

  1. I(v + v', v'') = I(v, v'') + I(v', v'') for all vectors v, v', and v'' in V.

  2. I(a v, v') = a I(v, v') for each scalar a and all vectors v and v' in V.

  3. I(v, v') = I(v', v) for all v and v' in V.

  4. I(v, v) > 0 for each v <> 0 in V.

• Definition. The length of a vector v in V relative to the inner product I is defined by the formula

  ||v|| = SQRT{I(v, v)} for v in V .

• Definition. The angle between two vectors in V relative to the inner product I is the unique angle (whose measure is) theta = angle (v, w) in the interval 0 <= theta <= pi satisfying the equation

  cos (theta) = {I(v, w)}/{ ||v|| ||w|| } for v, w in V .

• Definition. The orthogonal complement of a subspace W of V, relative to the inner product I on V, is the set W^{perp} that consists of all vectors v in V orthogonal to every vector in W, i.e., for which I(v, w) = 0 for all w in W.

• Theorem. If V is a vector space with a given inner product I, and W is any subspace of V, then the orthogonal complement of W has the following properties:

  1. W^{perp} is a linear subspace of V.

  2. The only point of V in both W and W^{perp} is the origin.

  3. The sum of the dimensions of W and W^{perp} is the dimension of V.

  4. Given a vector v in V there is a unique pair (w, w') of vectors with w in W and w' in W^{perp} such that v = w + w'.

  5. Every vector in W is perpendicular to every vector in W^{perp}.

Assignment for Wednesday, April 14

1. Let P_{3} be the vector space of polynomials of degree at most 3, and let phi be the linear map from P_{3} to itself that is defined by the formula

   (phi(f))(x) = INT[_{0}^{x} f'(t) d t ] ,

   where f' denotes the derivative of f. Find the matrix of phi with respect to the basis of P_{3} given by the powers of the variable.

2. Let P_{2} be the vector space of polynomials of degree at most 2. Define an inner product on P_{2} with the formula

   < f, g> = INT[_{0}^{1} f(t) g(t) dt ] .

   Find the orthogonal complement relative to this inner product of the subspace consisting of the constant polynomials.

3. Use the Gram-Schmidt process (§ 3.4 of the text) to make an orthonormal basis for P_{2}, relative to the inner product of the previous exercise containing the constant polynomial 1 beginning with the basis {1, t, t^{2}}.

4. Repeat the previous exercise using the inner product

   < f, g > = {1}/{2}INT[_{-1}^{1} f(t) g(t) dt ] .