91Ó°ÊÓ

Given that N is nilpotent, there exists a positive integer k such that N^k = 0. We will now prove the existence of a chain of N-invariant subspaces in V by induction. For the base case, let W_0 = {0}. Since N(W_0) is a subset of W_0, this confirms that W_0 is an N-invariant subspace. Assume that for some i ≥ 0, we have a chain of N-invariant subspaces: W_0 ⊂ W_1 ⊂ ... ⊂ W_i. Let x be any non-zero vector in W_i, and consider the set of vectors {x, N(x), N^2(x), ..., N^i(x)} (Note that N^(i+1)(x) = 0 since N is nilpotent). This set of vectors spans a subspace, denoted as U_i, that is invariant under N. Now, let W_(i+1) = W_i + U_i. Since W_i ⊆ W_(i+1) and U_i ⊆ W_(i+1), this shows that W_(i+1) is also an N-invariant subspace. By induction, we have successfully proven the existence of a chain of N-invariant subspaces in V.

For each of the N-invariant subspaces, W_i, we will select a basis and apply the Gram-Schmidt process to create an orthonormal basis spanned by the set {w_1, w_2, ..., w_i}: 1. Choose any non-zero vector w_1 in W_1 and normalize it: w_1 = w_1 / \|w_1\|. 2. For each successive vector, w_j, in W_j do the following: a. Subtract the projection of w_j onto the subspace spanned by the previous orthonormal vectors: \(w_j = w_j - \sum_{k=1}^{j-1} \langle w_j, w_k \rangle w_k\). b. Normalize the resulting vector: \(w_j = w_j/ \|w_j\|\). Repeat the process for all i up to the highest index of the chain of N-invariant subspaces.

With the orthonormal basis constructed in Step 2, we will express the matrix of N with respect to the basis {w_1, w_2, ..., w_n}. Since N is linear and respects the N-invariant subspaces, we can write N(w_j) as a linear combination of the preceding basis vectors: \(N(w_j) = \sum_{k=1}^{j} a_{k,j} w_k\). Now, the matrix representation of N has entries given by the coefficients a_{k, j} with respect to the orthonormal basis. Since k is always less than or equal to j in the linear combination, this ensures that the matrix of N is upper-triangular. Thus, we have successfully shown that there exists an orthonormal basis of V with respect to which the matrix of N is upper-triangular.