×
Get Full Access to Math - Textbook Survival Guide
Get Full Access to Math - Textbook Survival Guide
×

# Solutions for Chapter 6.6: Introduction to Homogeneous Coordinates (Optional)

## Full solutions for Elementary Linear Algebra with Applications | 9th Edition

ISBN: 9780132296540

Solutions for Chapter 6.6: Introduction to Homogeneous Coordinates (Optional)

Solutions for Chapter 6.6
4 5 0 327 Reviews
28
0
##### ISBN: 9780132296540

Chapter 6.6: Introduction to Homogeneous Coordinates (Optional) includes 23 full step-by-step solutions. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780132296540. Since 23 problems in chapter 6.6: Introduction to Homogeneous Coordinates (Optional) have been answered, more than 13598 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. This expansive textbook survival guide covers the following chapters and their solutions.

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

Tv = Av + Vo = linear transformation plus shift.

• Associative Law (AB)C = A(BC).

Parentheses can be removed to leave ABC.

• 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).

• Condition number

cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

• Cross product u xv in R3:

Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

• Distributive Law

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

• Eigenvalue A and eigenvector x.

Ax = AX with x#-O so det(A - AI) = o.

• Hankel matrix H.

Constant along each antidiagonal; hij depends on i + j.

• 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.

• Iterative method.

A sequence of steps intended to approach the desired solution.

• 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.

• lA-II = l/lAI and IATI = IAI.

The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.

• Network.

A directed graph that has constants Cl, ... , Cm associated with the edges.

• Pivot.

The diagonal entry (first nonzero) at the time when a row is used in elimination.

• Random matrix rand(n) or randn(n).

MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

• Reduced row echelon form R = rref(A).

Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

• Skew-symmetric matrix K.

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

• Symmetric factorizations A = LDLT and A = QAQT.

Signs in A = signs in D.

• Volume of box.

The rows (or the columns) of A generate a box with volume I det(A) I.

×