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# Solutions for Chapter 3.F: Duality

## Full solutions for Linear Algebra Done Right (Undergraduate Texts in Mathematics) | 3rd Edition

ISBN: 9783319110790

Solutions for Chapter 3.F: Duality

Solutions for Chapter 3.F
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##### ISBN: 9783319110790

Since 37 problems in chapter 3.F: Duality have been answered, more than 6106 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra Done Right (Undergraduate Texts in Mathematics), edition: 3. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra Done Right (Undergraduate Texts in Mathematics) was written by and is associated to the ISBN: 9783319110790. Chapter 3.F: Duality includes 37 full step-by-step solutions.

Key Math Terms and definitions covered in this textbook
• Affine transformation

Tv = Av + Vo = linear transformation plus shift.

• Block matrix.

A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

• Column picture of Ax = b.

The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

• Commuting matrices AB = BA.

If diagonalizable, they share n eigenvectors.

• Complex conjugate

z = a - ib for any complex number z = a + ib. Then zz = Iz12.

• 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.

• Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

Use AT for complex A.

• 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!).

• Hessenberg matrix H.

Triangular matrix with one extra nonzero adjacent diagonal.

• Independent vectors VI, .. " vk.

No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

• Inverse matrix A-I.

Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.

• 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.

• Minimal polynomial of A.

The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

• Multiplier eij.

The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

• Plane (or hyperplane) in Rn.

Vectors x with aT x = O. Plane is perpendicular to a =1= O.

• Projection p = a(aTblaTa) onto the line through a.

P = aaT laTa has rank l.

• Saddle point of I(x}, ... ,xn ).

A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

• Symmetric matrix A.

The transpose is AT = A, and aU = a ji. A-I is also symmetric.

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