Definition. A Euclidean vector space is a vector space with a given inner product.
In a Euclidean vector space it makes sense to speak of lengths and angles. However, the orthogonal linear maps are the only linear maps from a Euclidean vector space to itself that preserve lengths and angles.
Definition. If f is a linear map from a vector space V to itself, an f-invariant subspace of V is a linear subspace U of V for which f(u) is always in U whenever u is in U.
If the vector space is Euclidean and the linear map is either symmetric or orthogonal, then the orthogonal complement of an invariant subspace (if there is one) for a given linear map is also an invariant subspace for the same linear map.
A linear map of R^{2}, even when orthogonal or anti-symmetric, need not have invariant subspaces other than {0} and the whole plane. (See exercise 6 on the previous assignment.)
One seeks to understand the structure of a linear map from a vector space to itself by analyzing its behavior on its invariant subspaces to the extent that this is possible.
Show that the linear map f given by f(x) = T x where
| 1 | -1 |
| 0 | 1 |
has an invariant 1-dimensional subspace whose orthogonal complement is not f-invariant.
Let M be the 3 \times 3 orthogonal matrix
| 1 | 2 | 2 |
| 2 | 1 | -2 |
| 2 | -2 | 1 |
Find the invariant subspaces of the map x -> M x.
Hint. Look at f(1, 1, 0).
What can be said about the matrix of a linear map of an n-dimensional vector space relative to a basis v_{1}, ..., v_{n} if the span of the first r members v_{1}, ..., v_{r} is an invariant subspace for some r, 1 <= r < n? Suggestion. Begin by considering the cases n = 2, r=1, n = 3, r=1, and n = 3, r=2.