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# Solutions for Chapter 5.5: Inner Product Spaces

## Full solutions for Linear Algebra with Applications | 4th Edition

ISBN: 9780136009269

Solutions for Chapter 5.5: Inner Product Spaces

Solutions for Chapter 5.5
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##### ISBN: 9780136009269

This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 4. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009269. Chapter 5.5: Inner Product Spaces includes 35 full step-by-step solutions. Since 35 problems in chapter 5.5: Inner Product Spaces have been answered, more than 63828 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
• 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).

• Column space C (A) =

space of all combinations of the columns of A.

• Complete solution x = x p + Xn to Ax = b.

(Particular x p) + (x n in nullspace).

• Ellipse (or ellipsoid) x T Ax = 1.

A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad

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

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

• Fibonacci numbers

0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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

Use AT for complex A.

• Identity matrix I (or In).

Diagonal entries = 1, off-diagonal entries = 0.

• Jordan form 1 = M- 1 AM.

If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

• Left inverse A+.

If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.

• 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.

• Rank one matrix A = uvT f=. O.

Column and row spaces = lines cu and cv.

• Singular Value Decomposition

(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

• Skew-symmetric matrix K.

The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

• Solvable system Ax = b.

The right side b is in the column space of A.

• Special solutions to As = O.

One free variable is Si = 1, other free variables = o.

• Standard basis for Rn.

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

• Symmetric matrix A.

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

• Vector space V.

Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.