Reference no: EM133343383
Question 1. Find an LU-factorization for
Question 2. Let a, b, c, d , e, f ∈ C with ace ≠ 0 Use reduction to row echelon form to show that
is invertible and find its inverse in terms of a, b, c, d , e, f . Why do we require ace ≠ 0?
Question 3. Let S = {(3, -2, 2, 1), (5, -3, 4, 2), (-4, 4, 0, 0), (-6, 5, -2, -1)} and let W = span(S) be a subspace of R4.
(a) Find a subset B of S such that B is a basis for W.
(b) Find a basis C for R4 such that B is a subset of C.
Question 4. Prove by induction that for all k ≥ 2, the product of k lower triangular matrices is lower triangular. That is, if L1, . . . , Lk ∈ Fn×n are lower triangular matrices, then L1,L2 . . . , Lk is lower triangular.
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