- Here are three vectors in ℝ
^{4}:^{T},^{T},^{T}. Find the three dimensional volume of the parallelepiped determined by these three vectors. - Here are two vectors in ℝ
^{4}:^{T},^{T}. Find the volume of the parallelepiped determined by these two vectors. - Here are three vectors in ℝ
^{2}:^{T},^{T},^{T}. Find the three dimensional volume of the parallelepiped determined by these three vectors. Recall that from the above theorem, this should equal 0. - Find the equation of the plane through the three points ,,.
- Let T map a vector space V to itself. Explain why T is one to one if and only if T is onto. It is in the text, but do it again in your own words.
- ↑Let all matrices be complex with complex field of scalars and let A be an n × n
matrix and B a m × m matrix while X will be an n × m matrix. The problem is
to consider solutions to Sylvester’s equation. Solve the following equation for
X
where C is an arbitrary n × m matrix. Show there exists a unique solution if and only if σ

∩ σ= ∅. Hint: If qis a polynomial, show first that if AX − XB = 0, then qX −Xq= 0. Next define the linear map T which maps the n×m matrices to the n × m matrices as follows.Show that the only solution to TX = 0 is X = 0 so that T is one to one if and only if σ

∩σ= ∅. Do this by using the first part for qthe characteristic polynomial for B and then use the Cayley Hamilton theorem. Explain why q^{−1}exists if and only if the condition σ∩ σ= ∅. - Compare Definition 11.8.2 with the Binet Cauchy theorem, Theorem 3.3.14. What is the geometric meaning of the Binet Cauchy theorem in this context?
- For W a subspace of V, W is said to have a complementary subspace [15] W
^{′}if W ⊕W^{′}= V. Suppose that both W,W^{′}are invariant with respect to A ∈ℒ. Show that for any polynomial f, if fx ∈ W, then there exists w ∈ W such that fx = fw. A subspace W is called A admissible if it is A invariant and the condition of this problem holds. - ↑ Return to Theorem 9.3.5 about the existence of a basis β = for V where A ∈ℒ. Adapt the statement and proof to show that if W is A admissible, then it has a complementary subspace which is also A invariant. Hint:
The modified version of the theorem is: Suppose A ∈ℒ

and the minimal polynomial of A is ϕ^{m}where ϕis a monic irreducible polynomial. Also suppose that W is an A admissible subspace. Then there exists a basis for V which is of the form β =whereis a basis of W. Thus spanis the A invariant complementary subspace for W. You may want to use the fact that ϕ∩ W = ϕwhich follows easily because W is A admissible. Then use this fact to show that ϕis also A admissible. - Let U,H be finite dimensional inner product spaces. (More generally, complete inner product
spaces.) Let A be a linear map from U to H. Thus AU is a subspace of H. For g ∈ AU,
define A
^{−1}g to be the unique element ofwhich is closest to 0. Then define_{AU}≡_{U}. Show that this is a well defined inner product. Let U,H be finite dimensional inner product spaces. (More generally, complete inner product spaces.) Let A be a linear map from U to H. Thus AU is a subspace of H. For g ∈ AU, define A^{−1}g to be the unique element ofwhich is closest to 0. Then define_{AU}≡_{U}. Show that this is a well defined inner product and that if A is one to one, then_{AU}=_{U}and_{AU}=_{U}.

Download PDFView PDF