- 6.6.1: Refer to Example 8. (a) Determine the matrix M in homogeneous fonn ...
- 6.6.2: Let the columns of 2 4 J 4 be the homogeneous form of the coordinat...
- 6.6.3: Let the columns of 2 4 3 4 ~] be the homogeneous fonn of the coordi...
- 6.6.4: A plane figure S is to be translated by I = [ -~ ] and then the res...
- 6.6.5: Let A be the 3 x 3 matrix in homogeneous form that reflects a plane...
- 6.6.6: LeI A be the 3 x 3 matrix in homogeneous fonllthat translates a pla...
- 6.6.7: Detem)jne the malrix in homogcncou$ fonn Ihal produced Ihe image of...
- 6.6.8: Determine lhe matrix in homogcneou$ fonn Ihal produced the image of...
- 6.6.9: Determine [he matrix in homogeneous fonn Ihal pro d uced the image ...
- 6.6.10: Determine the matrix in hor.lOgeneous fonn that pro duced the image...
- 6.6.11: Determine the matrix in homogeneous fonn that produced the image of...
- 6.6.12: Determine the matrix in hor.lOgeneous fonn that produced the image ...
- 6.6.13: The semicircle depicted as the original figure in Exercise [2 is to...
- 6.6.14: Sweeps In calculus a surface of revol ution is generated by rotming...
- 6.6.15: Let L: P3 -+ P3 be defined by L (lIf3 + bf2 + cf + d ) = 3ar 2 + 2b...
- 6.6.16: Consider R" as an inner prod\lct space with the standard mner produ...
- 6.6.17: Let LI : V -+ V and L2: V -+ V be linear transformations on a vecto...
- 6.6.18: Let u and v be nonzero vectors in R". In Section 5.3 we defined the...
- 6.6.19: Let L: N" -+ R" be a linear operator that preserves inner producL~ ...
- 6.6.20: Let L: V -;. IV be ;I linear transformation. If (VI. V2 .. vd spans...
- 6.6.21: Let V be an II-dimensional vector space and S (VI.V2 ..... V,,) a b...
- 6.6.22: Let V be an II-dimensional vector space. The \ector space of all li...
- 6.6.23: If A and B are nonsingular, show that AB and BA are similar.
Solutions for Chapter 6.6: Introduction to Homogeneous Coordinates (Optional)
Full solutions for Elementary Linear Algebra with Applications | 9th Edition
Tv = Av + Vo = linear transformation plus shift.
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
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).
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 II· 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].
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.
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.
A directed graph that has constants Cl, ... , Cm associated with the edges.
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.