Definition. If I is an inner product on a finite-dimensional vector space V and if v = (v_{1}, ..., v_{n}) is a basis of V, then the matrix of I relative to v is the matrix (I(v_{i}, v_{j})).
Note that the notion of the matrix of an inner product relative to a basis is a notion that is entirely separate from the notion of the matrix of a linear map relative to a basis.
Proposition. If S is an n \times n symmetric matrix, then the formula
where x^{*} denotes the row that is the transpose of the column x, defines a function of two point variables x and y in R^{n} that satisfies all of the rules for an inner product except the rule specifying that all ``lengths'' I(x, x) must be positive for all points x except the origin. A matrix for which the last condition holds is called a positive-definite symmetric matrix.
Proposition. If I is an inner product on a finite-dimensional vector space V and if v = (v_{1}, ..., v_{n}) and w = (w_{1}, ..., w_{n}) are bases of V with w = v A, where A is the matrix for change of basis, then the matrix S of I relative to the basis v and the matrix T of I relative to the basis w are related by the formula
Show that the determinant of the 2 \times 2 matrix
| 1 | 0 |
| 0 | 1 |
| a | b |
| c | d |
is an element of the vector space P_{2} of polynomials of degree at most 2 when viewed as a function of t for given values of a, b, c, and d.
Let I be the inner product in R^{3} that is defined by
What is the matrix of I relative to the standard basis of R^{3} ?
What is the matrix of I relative to the basis of R^{3} formed by the columns of the matrix
| 1 | 2 | 2 |
| 2 | 1 | -2 |
| 2 | -2 | 1 |
In second semester calculus one learns that the second degree curve in R^{2} given by the equation
can always be put in ``standard form'' when B^{2} - 4 A C <> 0 by rotating the coordinate axes. What are the possible standard forms, and how might this be re-stated in the language of linear algebra?