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# Solutions for Chapter 2-5: Algebraic Proof

## Full solutions for Geometry | 1st Edition

ISBN: 9780030923456

Solutions for Chapter 2-5: Algebraic Proof

Solutions for Chapter 2-5
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##### ISBN: 9780030923456

This textbook survival guide was created for the textbook: Geometry, edition: 1. Geometry was written by and is associated to the ISBN: 9780030923456. Since 50 problems in chapter 2-5: Algebraic Proof have been answered, more than 42124 students have viewed full step-by-step solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 2-5: Algebraic Proof includes 50 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• Cayley-Hamilton Theorem.

peA) = det(A - AI) has peA) = zero matrix.

• Cofactor Cij.

Remove row i and column j; multiply the determinant by (-I)i + j •

• Commuting matrices AB = BA.

If diagonalizable, they share n eigenvectors.

• Companion matrix.

Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).

• Diagonalizable matrix A.

Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

• Distributive Law

A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

• Echelon matrix U.

The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

• Exponential eAt = I + At + (At)2 12! + ...

has derivative AeAt; eAt u(O) solves u' = Au.

• Free variable Xi.

Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

• Gauss-Jordan method.

Invert A by row operations on [A I] to reach [I A-I].

• Hessenberg matrix H.

Triangular matrix with one extra nonzero adjacent diagonal.

• Indefinite matrix.

A symmetric matrix with eigenvalues of both signs (+ and - ).

• Kronecker product (tensor product) A ® B.

Blocks aij B, eigenvalues Ap(A)Aq(B).

• Krylov subspace Kj(A, b).

The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

• Linear transformation T.

Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

• Nullspace N (A)

= All solutions to Ax = O. Dimension n - r = (# columns) - rank.

• Orthonormal vectors q 1 , ... , q n·

Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

• Spanning set.

Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

• Standard basis for Rn.

Columns of n by n identity matrix (written i ,j ,k in R3).

• Vector v in Rn.

Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

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