 67.67.1: In Exercises 1 and 2, determine whether the function is a linear tr...
 67.67.2: In Exercises 1 and 2, determine whether the function is a linear tr...
 67.67.3: Let T: RnRm be the linear transformation defined by T(v) = Av, wher...
 67.67.4: Let T: R2R3 be the linear transformation defined by T(v) = Av, wher...
 67.67.5: Find the kernel of the linear transformationT: R4R4, T(x1, x2, x3, ...
 67.67.6: Let T: R4R2 be the linear transformation defined by T(v) = Av, wher...
 67.67.7: In Exercises 710, find the standard matrix for the linear transform...
 67.67.8: In Exercises 710, find the standard matrix for the linear transform...
 67.67.9: In Exercises 710, find the standard matrix for the linear transform...
 67.67.10: In Exercises 710, find the standard matrix for the linear transform...
 67.67.11: Find the standard matrix A for the linear transformation projvu: R2...
 67.67.12: Let T: R2R2 be the linear transformation defined by a counterclockw...
 67.67.13: In Exercises 13 and 14, find the standard matrices for T = T2 T1 an...
 67.67.14: In Exercises 13 and 14, find the standard matrices for T = T2 T1 an...
 67.67.15: Find the inverse of the linear transformation T: R2R2 defined byT(x...
 67.67.16: Determine whether the linear transformation T: R3R3 defined byT(x1,...
 67.67.17: Find the matrix of the linear transformation T(x, y) = (y, 2x, x + ...
 67.67.18: Let B = {(1, 0), (0, 1)} and B = {(1, 1), (1, 2)} be bases for R2.(...
 67.67.19: In Exercises 1922, find the eigenvalues and the corresponding eigen...
 67.67.20: In Exercises 1922, find the eigenvalues and the corresponding eigen...
 67.67.21: In Exercises 1922, find the eigenvalues and the corresponding eigen...
 67.67.22: In Exercises 1922, find the eigenvalues and the corresponding eigen...
 67.67.23: In Exercises 23 and 24, find a nonsingular matrix P such that P1AP ...
 67.67.24: In Exercises 23 and 24, find a nonsingular matrix P such that P1AP ...
 67.67.25: Find a basis B for R3 such that the matrix for the linear transform...
 67.67.26: Find an orthogonal matrix P such that PTAP diagonalizes the symmetr...
 67.67.27: Use the GramSchmidt orthonormalization process to find an orthogon...
 67.67.28: Solve the system of differential equations.y1 = y1y2 = 9y2
 67.67.29: Find the matrix of the quadratic form associated with the quadratic...
 67.67.30: A population has the following characteristics.(a) A total of 80% o...
 67.67.31: Define an orthogonal matrix
 67.67.32: Prove that if A is similar to B and A is diagonalizable, then B is ...
Solutions for Chapter 67: Cumulative Test
Full solutions for Elementary Linear Algebra  8th Edition
ISBN: 9781305658004
Solutions for Chapter 67: Cumulative Test
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 32 problems in chapter 67: Cumulative Test have been answered, more than 44312 students have viewed full stepbystep solutions from this chapter. Elementary Linear Algebra was written by and is associated to the ISBN: 9781305658004. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 8. Chapter 67: Cumulative Test includes 32 full stepbystep solutions.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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).

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

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 matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

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

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

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

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.

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.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

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

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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.