7.8 Problems

Problem 1. Let u₁,…,u_k ∈ℝⁿ be a set of linearly independent and pairwise orthogonal vectors. Show that the linear transformation

is an orthogonal projection.

Problem 2. Let u₁,…,u_k ∈ℝⁿ be a set of linearly independent vectors, and define the linear transformation

Is this a projection? (Hint: Study the special case k = 2 and ℝ³. You can visualize this if needed.)

Problem 3. Let V be an inner product space and P : V →V be an orthogonal projection. Show that I −P is an orthogonal projection as well, and

holds.

Problem 4. Let A,B ∈ℝ^n×n be two square matrix that are written in the block matrix form

where A_1,1,B_1,1 ∈ℝ^k×k, A_1,2,B_1,2 ∈ℝ^k×l, A_2,1,B_2,1 ∈ℝ^l×k, and A_2,2,B_2,2 ∈ℝ^l×l.

Show that

Problem 5. Let A ∈ℝ^2×2 be the square matrix defined by

(a) Show that the two eigenvalues of A are

...