4.6 Problems

Problem 1. Show that if A ∈ℝ^n×n is an invertible matrix, then

Problem 2. Let R_α be the two-dimensional rotation matrix defined by

Show that R_αR_β = R_α+β.

Problem 3. Let A = (a_i,j)_i,j=1ⁿ ∈ℝ^n×n be a matrix and let D ∈ℝ^n×n be a diagonal matrix defined by

where all of its elements are zero outside the diagonal. Show that

and

Problem 4. Let ∥⋅∥ be a norm on ℝⁿ, and let A ∈ℝ^n×n be an arbitrary matrix.

Show that A is invertible if and only if the function

is a norm on ℝⁿ.

Problem 5. Let U be a normed space and f : U →U be a linear transformation.

If

is a norm, is f necessarily invertible?

Hint: Consider the vector space ℝ[x] with the norm

and the linear transformation f : p(x)→xp(x).

Problem 6. Let ⟨⋅,⋅,⟩ be an inner product on ℝⁿ. Show that there is a matrix A...