Introduction to Linear Algebra 4th Edition solutions

Introduction to Linear Algebra 4
Questions 1-7 are about the rules for transpose matrices.

Introduction to Linear Algebra 4
Questions 1-7 are about the rules for transpose matrices.

Introduction to Linear Algebra 4
Questions 1-7 are about the rules for transpose matrices.

Introduction to Linear Algebra 4
Questions 1-7 are about the rules for transpose matrices.

Introduction to Linear Algebra 4
Questions 1-7 are about the rules for transpose matrices.

Introduction to Linear Algebra 4
Questions 1-7 are about the rules for transpose matrices.

Introduction to Linear Algebra 4
Questions 1-7 are about the rules for transpose matrices.

Introduction to Linear Algebra 4
Questions 8-15 are about permutation matrices.

Introduction to Linear Algebra 4
Questions 8-15 are about permutation matrices.

Introduction to Linear Algebra 4
Questions 8-15 are about permutation matrices.

Introduction to Linear Algebra 4
Questions 8-15 are about permutation matrices.

Introduction to Linear Algebra 4
Questions 8-15 are about permutation matrices.

Introduction to Linear Algebra 4
Questions 8-15 are about permutation matrices.

Introduction to Linear Algebra 4
Questions 8-15 are about permutation matrices.

Introduction to Linear Algebra 4
Questions 8-15 are about permutation matrices.

Introduction to Linear Algebra 4
Questions 16-21 are about symmetric matrices and their factorizations.

Introduction to Linear Algebra 4
Questions 16-21 are about symmetric matrices and their factorizations.

Introduction to Linear Algebra 4
Questions 16-21 are about symmetric matrices and their factorizations.

Introduction to Linear Algebra 4
Questions 16-21 are about symmetric matrices and their factorizations.

Introduction to Linear Algebra 4
Questions 16-21 are about symmetric matrices and their factorizations.

Introduction to Linear Algebra 4
Questions 16-21 are about symmetric matrices and their factorizations.

Introduction to Linear Algebra 4
Questions 22-24 are aboutthe factorizations PA = L U and A = LIP 1 U 1-

Introduction to Linear Algebra 4
Questions 22-24 are aboutthe factorizations PA = L U and A = LIP 1 U 1-

Introduction to Linear Algebra 4
Questions 22-24 are aboutthe factorizations PA = L U and A = LIP 1 U 1-

Introduction to Linear Algebra 4
Extend the slu code in Section 2.6 to a code splu that factors P A into L U .

Introduction to Linear Algebra 4
Prove that the identity matrix cannot be the product of three row exchanges (or five). It can be the product of two exchanges (or four).

Introduction to Linear Algebra 4
(a) Choose E21 to remove the 3 below the first pivot. Then mUltiply E21AEi1 to remove both 3 's: 3 0] 11 4 4 9 [ 1 0 0] is going toward D = 0 2 0 . o 0 1 (b) Choose E32 to remove the 4 below the second pivot. Then A is reduced to D by E32E21AEi1 Ej2 = D. Invert the E's to find L in A = LDLT.

Introduction to Linear Algebra 4
If every row of a 4 by 4 matrix contains the numbers 0,1,2,3 in some order, can the matrix be symmetric?

Introduction to Linear Algebra 4
Prove that no '.reordering of rows and reordering of columns can transpose a typical matrix. (Watch the diagonal entries.)

Introduction to Linear Algebra 4
Wires go between Boston, Chicago, and Seattle. Those cities are at voltages XB, Xc, Xs. With unit resistances between cities, the currents between cities are in y: y = Ax is [~~~] = [~ -~ -~] [~~] . YBS 1 0 -1 Xs (a) Find the total currents AT y out ofthe three cities. (b) Verify that (Ax) T y agrees

Introduction to Linear Algebra 4
Producing Xl trucks and X2 planes needs Xl + 50X2 tons of steel, 40XI + 1000x2 pounds of rubber, and 2XI + 50X2 months oflabor. If the unit costs YI, Y2, Y3 are $700 per ton, $3 per pound, and $3000 per month, what are the values of one truck and one plane? Those are the components of AT y.

Introduction to Linear Algebra 4
Ax gives the amounts of steel, rubber, and labor to produce x in Problem 31. Find A. Then Ax y is the of inputs while x AT Y is the value of __

Introduction to Linear Algebra 4
The matrix P that mUltiplies (x, y, z) to give (z, X, y) is also a rotation matrix. Find P and P 3 . The rotation axis a = (1, 1, 1) doesn't move, it equals P a. What is the angle of rotation from v = (2,3, -5) to Pv = (-5,2, 3)?

Introduction to Linear Algebra 4
Write A = [l;] as the product E H of an elementary row operation matrix E and a symmetric matrix H.

Introduction to Linear Algebra 4
Here is a new factorization of A into triangular (with 1 's) times symmetric: Start from A = LDU. Then A = L(UT)-l times UT DU. Why is L(UT)-1 triangular? Its diagonal is alII's. Why is UT DU symmetric?

Introduction to Linear Algebra 4
A group of matrices includes A B and A -1 if it includes A and B. "Products and inverses stay in the group." Which of these sets are groups? Lower triangular matrices L with 1 's on the diagonal, symmetric matrices S, positive matrices M, diagonal invertible matrices D, permutation matrices P, matric

Introduction to Linear Algebra 4
A square northwest matrix B is zero in the southeast comer, below the antidiagonal that connects (1, n) to (n, 1). Will BT and B2 be northwest matrices? Will B-1 be northwest or southeast? What is the shape of BC = northwest times southeast?

Introduction to Linear Algebra 4
If you take powers of a permutation matrix, why is some pk eventually equal to I? Find a 5 by 5 permutatioh P so that the smallest power to equal I is P 6 .

Introduction to Linear Algebra 4
(a) Write down any 3 by 3 matrix A. Split A into B + C where B = BT is symmetric and C = -CT is anti-symmetric. (b) Find formulas for Band C involving A and AT. We want A = B + C with B = BT and C = -CT.

Introduction to Linear Algebra 4
Suppose QT equals Q-l (transpose equals inverse, so QT Q = l). (a) Show that the columns ql, ... , qn are unit vectors: Ilq dl 2 = 1. (b) Show that every two columns of Q are perpendicular: q T q 2 = O. (c) Find a 2 by 2 example with first entry qll = cos e.