 4.11.1: Find the standard matrix for the operator that maps a point into (a...
 4.11.2: For each part of Exercise 1, use the matrix you have obtained to co...
 4.11.3: Find the standard matrix for the operator that maps a point into (a...
 4.11.4: For each part of Exercise 3, use the matrix you have obtained to co...
 4.11.5: Find the standard matrix for the operator that (a) rotates each vec...
 4.11.6: Sketch the image of the rectangle with vertices , , , and under (a)...
 4.11.7: Sketch the image of the square with vertices , and under multiplica...
 4.11.8: Find the matrix that rotates a point about the origin (a) 45 (b) 90...
 4.11.9: Find the matrix that shears by (a) a factor of in the ydirection. ...
 4.11.10: Find the matrix that compresses or expands by (a) a factor of in th...
 4.11.11: In each part, describe the geometric effect of multiplication by A....
 4.11.12: In each part, express the matrix as a product of elementary matrice...
 4.11.13: In each part, find a single matrix that performs the indicated succ...
 4.11.14: In each part, find a single matrix that performs the indicated succ...
 4.11.15: Use matrix inversion to show the following. (a) The inverse transfo...
 4.11.16: Find an equation of the image of the line under multiplication by
 4.11.17: In parts (a) through (e), find an equation of the image of the line...
 4.11.18: Find the matrix for a shear in the xdirection that transforms the ...
 4.11.19: (a) Show that multiplication by maps each point in the plane onto t...
 4.11.20: Prove part (a) of Theorem 4.11.3. [Hint: A line in the plane has an...
 4.11.21: Use the hint in Exercise 20 to prove parts (b) and (c) of Theorem 4...
 4.11.22: In each part of the accompanying figure, find the standard matrix f...
 4.11.23: In the shear in the xydirection with factor k is the matrix transf...
 4.11.a: In parts (a)(g) determine whether the statement is true or false, a...
 4.11.b: In parts (a)(g) determine whether the statement is true or false, a...
 4.11.c: In parts (a)(g) determine whether the statement is true or false, a...
 4.11.d: In parts (a)(g) determine whether the statement is true or false, a...
 4.11.e: In parts (a)(g) determine whether the statement is true or false, a...
 4.11.f: In parts (a)(g) determine whether the statement is true or false, a...
 4.11.g: In parts (a)(g) determine whether the statement is true or false, a...
Solutions for Chapter 4.11: Geometry of Matrix Operators on
Full solutions for Elementary Linear Algebra: Applications Version  10th Edition
ISBN: 9780470432051
Solutions for Chapter 4.11: Geometry of Matrix Operators on
Get Full SolutionsSince 30 problems in chapter 4.11: Geometry of Matrix Operators on have been answered, more than 14322 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra: Applications Version, edition: 10. Elementary Linear Algebra: Applications Version was written by and is associated to the ISBN: 9780470432051. Chapter 4.11: Geometry of Matrix Operators on includes 30 full stepbystep solutions.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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 IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

lAII = 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.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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

Outer product uv T
= column times row = rank one matrix.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).