 1.7.1: Let I: /?! Jo /?2 be the matrix transformation defined by I(v) = A...
 1.7.2: Let R be the rectangle with vertices (I. I). (I. 4). (3. I). and (3...
 1.7.3: A shcar in the ydirection is the matrix transformation I: /?! .......
 1.7.4: The matrix transformation j: /?! ...,. /?! defined by lev) = Av. wh...
 1.7.5: The matrix transformation I: /?2 ...... /?2 defined by I(v) = Al. w...
 1.7.6: The matrix transformation I: /?2 ...... /?2 defined by I(v) = Al. w...
 1.7.7: Let T be the triangle with ven ices (5. 0), (0. 3), and (2.  I). F...
 1.7.8: Let T be the triangle with vertices ( I . I). (3.  3), and (2.  ...
 1.7.9: Let I be the counterclockwise rotation through 600 If T is the tria...
 1.7.10: Let II be reflection with respect to the yaxis and let h be counte...
 1.7.11: Let A be the smgular matnx . [' 2 '] 4 and leI T be the triangle de...
 1.7.12: LeI f be [he malrix transformation defined in Example 5. Find and s...
 1.7.13: Let f: R2 I' R2 be the matrix lr3nSfomlation defined by /(11) = ...
 1.7.14: 111 .um:is/!,\" /4 (l1ll115, In f l. h. /J. and /4 be the/ollowillK...
 1.7.15: 112 .um:is/!,\" /4 (l1ll115, In f l. h. /J. and /4 be the/ollowillK...
 1.7.16: Refer to the discussion following Example 3 to develop the double a...
 1.7.17: Use a procedure similar to the one discussed after Example 3 to dev...
 1.7.18: Define a triangle T by identifying its vertices and sketch It on pa...
 1.7.19: Consider the triangle T defined in Exercise 18. Record r on paper.
 1.7.20: Consider the unit sq uare S and record S on paper. (a) Reflect S ab...
 1.7.21: If your complller graphics software allows you to select any 2 x 2 ...
 1.7.22: If your software includes access to a computer algebra system (CAS)...
 1.7.23: Ie yuur ~unwan: indmks an:ess tu a (;UlTlpUler algebra system (CAS)...
 1.7.24: If your software includes access to a computer algebra system (CAS)...
Solutions for Chapter 1.7: Computer Graphics (Optional)
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780471669593
Solutions for Chapter 1.7: Computer Graphics (Optional)
Get Full SolutionsElementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780471669593. Since 24 problems in chapter 1.7: Computer Graphics (Optional) have been answered, more than 9187 students have viewed full stepbystep solutions from this chapter. Chapter 1.7: Computer Graphics (Optional) includes 24 full stepbystep solutions. 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.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

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.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.