Definition. An orthogonal matrix is an invertible matrix for which the inverse of the matrix is equal to the transpose of the matrix.
Proposition. An n \times n matrix is an orthogonal matrix if and only if its columns are mutually perpendicular vectors of length 1 relative to the standard inner product in R^{n}.
Theorem. A linear map from R^{n} to itself is distance-preserving (relative to the standard inner product) if and only if its matrix (relative to the standard basis) is an orthogonal matrix.
Definition. If V is a vector space with given inner product I, a linear map f from V to itself is said to be an orthogonal linear map if for all v, w in V one has I(f(v), f(w)) = I(v, w).
Theorem. If V is a given vector space, v_{1}, ..., v_{n} an orthonormal basis of V relative to a given inner product on V, and f a linear map from V to V, then f is an orthgonal linear map relative to the inner product if and only if the matrix of f with respect to v_{1}, ..., v_{n} is an orthogonal matrix.
Show that the formula I(x, y) = 5 x_{1} y_{1} - 2(x_{1} y_{2} + x_{2} y_{1}) + x_{2} y_{2} defines an inner product on R^{2}.
What is the length of the vector (a, b) relative to the inner product in the preceding exercise?
What is the orthogonal complement in R^{2} of the line 2 x + y = 0 relative to:
the standard inner product?
the inner product I(x, y) = 5 x_{1} y_{1} - 2(x_{1} y_{2} + x_{2} y_{1}) + x_{2} y_{2} ?
Find the orthogonal complement relative to the standard inner product in R^{3} of the following linear subspaces:
The plane 2 x - y + 3 z = 0.
The line given by t (-1, 2, 5), where t is a parameter.
The line of points satisfying the equations 2 x + y - z = x - 3 y + 2 z = 0.
Let f be the linear map from R^{2} to itself given by f(x) = M x where M is the matrix
| 3/5 | 4/5 |
| 4/5 | -3/5 |
Find the matrix of f relative to the orthonormal basis { (2/SQRT{5}, 1/SQRT{5}), (-1/SQRT{5}, 2/SQRT{5}) } .
Show that the inverse and the transpose of an orthogonal matrix are also orthogonal matrices.
Show that the product of two orthogonal matrices is an orthogonal matrix.
Show that the determinant of an orthogonal matrix must be {+/-} 1.