Introduction to Linear Algebra 4th Edition solutions

Introduction to Linear Algebra 4
Problems 1-14 compute the factorization A = LV (and also A = LDV).

Introduction to Linear Algebra 4
Problems 1-14 compute the factorization A = LV (and also A = LDV).

Introduction to Linear Algebra 4
Problems 1-14 compute the factorization A = LV (and also A = LDV).

Introduction to Linear Algebra 4
Problems 1-14 compute the factorization A = LV (and also A = LDV).

Introduction to Linear Algebra 4
Problems 1-14 compute the factorization A = LV (and also A = LDV).

Introduction to Linear Algebra 4
Problems 1-14 compute the factorization A = LV (and also A = LDV).

Introduction to Linear Algebra 4
Problems 1-14 compute the factorization A = LV (and also A = LDV).

Introduction to Linear Algebra 4
Problems 1-14 compute the factorization A = LV (and also A = LDV).

Introduction to Linear Algebra 4
Problems 1-14 compute the factorization A = LV (and also A = LDV).

Introduction to Linear Algebra 4
Problems 1-14 compute the factorization A = LV (and also A = LDV).

Introduction to Linear Algebra 4
Problems 1-14 compute the factorization A = LV (and also A = LDV).

Introduction to Linear Algebra 4
Problems 1-14 compute the factorization A = LV (and also A = LDV).

Introduction to Linear Algebra 4
Problems 1-14 compute the factorization A = LV (and also A = LDV).

Introduction to Linear Algebra 4
Problems 1-14 compute the factorization A = LV (and also A = LDV).

Introduction to Linear Algebra 4
Problems 15-16 use Land U (without needing A) to solve Ax = b.

Introduction to Linear Algebra 4
Problems 15-16 use Land U (without needing A) to solve Ax = b.

Introduction to Linear Algebra 4
(a) When you apply the usual elimination steps to L, what matrix do you reach? L = [e!1 ~ ~]. e31 e32 I (b) When you apply the same steps to I, what matrix do you get? (c) When you apply the same steps to LU, what matrix do you get?

Introduction to Linear Algebra 4
If A = LDU and also A = Ll Dl U1 with all factors invertible, then L = Ll and D = DI and U = Ul. "The three/actors are unique." 19 20 Derive the equation Ll1 LD = Dl Ul U- 1 Are the two sides triangular or diagonal? Deduce L = Ll and U = Ul (they all have diagonall 's). Then D = Dl.

Introduction to Linear Algebra 4
Tridiagonal matrices have zero entries except on the main diagonal and the two adjacent diagonals. Factor these into A = L U and A = L D L T : [ a a 0] and A = a a + b b . o b b +c

Introduction to Linear Algebra 4
When T is tridiagonal, its Land U factors have only two nonzero diagonals. How would you take advant~ge of knowing the zeros in T, in a code for Gaussian elimination? Find Land U. \

Introduction to Linear Algebra 4
If A and B have nonzeros in the positions marked by x, which zeros (marked by 0) stay zero in their factors Land U?

Introduction to Linear Algebra 4
Suppose you eliminate upwards (almost unheard ot). Use the last row to produce zeros in the last column (the pivot is 1). Then use the second row to produce zero above the second pivot. Find the factors in the unusual order A = V L. Upper times lower

Introduction to Linear Algebra 4
Easy but important. If A has pivots 5, 9, 3 with no row exchanges, what are the pivots for the upper left 2 by 2 submatrix A2 (without row 3 and column 3)?

Introduction to Linear Algebra 4
Which invertible matrices allow A = LV (elimination without row exchanges)? Good question! Look at each of the square upper left submatrices of A. All upper left k by k submatrices Ak must be invertible (sizes k = 1, ... , n). Explain that answer: Ak factors into __ because LV = [;k ~] [6

Introduction to Linear Algebra 4
For the 6 by 6 second difference constant-diagonal matrix K, put the pivots and multipliers into K = LV. (L and V will have only two nonzero diagonals, because K has three.) Find a formula for the i, j entry of L -I, by software like MATLAB using inv(L) or by looking for a nice pattern.

Introduction to Linear Algebra 4
f you print K- 1 , ;it doesn't look so good. But if you print 7 K- 1 (when K is 6 by 6), that matrix looks wonderful. Write down 7 K- 1 by hand, following this pattern: 1 Row 1 and column 1 are (6,5,4,3,2,1). 2 On and above the main diagonal, row i is i times row 1. 3 On and below the main diagonal,