Question

Theorem Three.VI.2.7 describes a change of basis from any basis B = (v₁, ..., vₙ) to one that is orthogonal K = (u₁, ..., uₙ). Consider the change of basis matrix RepB. (id).

(a) Prove that the matrix Rep. (id) changing bases in the opposite direction to that of the theorem has an upper triangular shape—all of its entries below the main diagonal are zeros.

Answer 1

To prove that the matrix Rep. (id) changing bases in the opposite direction to that of Theorem Three.VI.2.7 has an upper triangular shape with all entries below the main diagonal being zeros, we can use induction.

Step-by-step explanation:

To prove that the matrix Rep. (id) changing bases in the opposite direction to that of Theorem Three.VI.2.7 has an upper triangular shape with all entries below the main diagonal being zeros, we can use induction.

1. Start with the case when the dimension n = 1. In this case, there is only one basis vector and the change of basis matrix Rep. (id) is just the identity matrix, which is an upper triangular matrix with all entries below the main diagonal being zeros.

2. Assume that the statement is true for n = k, where k is a positive integer. That is, the change of basis matrix Rep. (id) has an upper triangular shape with all entries below the main diagonal being zeros for a basis of dimension k.

3. Now consider the case when the dimension is n = k + 1. Let (v₁, ..., vₖ, vₖ₊₁) be a basis of dimension k + 1. Using the definition of the change of basis matrix, we can write Rep. (id) = [RepB. (id) vₖ+₁], where RepB. (id) is the change of basis matrix from the basis (v₁, ..., vₖ) to an orthogonal basis, and vₖ+₁ is the k + 1th basis vector.

4. Since RepB. (id) is an upper triangular matrix with all entries below the main diagonal being zeros (by the induction assumption), and vₖ+₁ is an orthogonal vector to the span of (v₁, ..., vₖ), the product [RepB. (id) vₖ+₁] is also an upper triangular matrix with all entries below the main diagonal being zeros.

5. Hence, by induction, the matrix Rep. (id) changing bases in the opposite direction to that of Theorem Three.VI.2.7 has an upper triangular shape with all entries below the main diagonal being zeros for bases of any dimension.