 5.1: If T is a linear transformation from Rn to Rn such that T (e1), T (...
 5.2: If A is an invertible matrix, then the equation (AT )1 = (A1)T must...
 5.3: If matrix A is orthogonal, then matrix A2 must be orthogonal as well.
 5.4: The equation (AB)T = AT BT holds for all n n matrices A and B.
 5.5: If A and B are symmetric n n matrices, then A + B must be symmetric...
 5.6: If matrices A and S are orthogonal, then S1AS is orthogonal as well.
 5.7: . All nonzero symmetric matrices are invertible.
 5.8: If A is an n n matrix such that AAT = In, then A must be an orthogo...
 5.9: If u is a unit vector in Rn, and L = span(u), then projL (x) = (x u...
 5.10: . If A is a symmetric matrix, then 7A must be symmetric as well.
 5.11: If x and y are two vectors in Rn, then the equation x + y2 = x2 + y...
 5.12: The equation det(AT ) = det(A) holds for all 2 2 matrices A
 5.13: If matrix A is orthogonal, then AT must be orthogonal as well.
 5.14: If A and B are symmetric nn matrices, then AB must be symmetric as ...
 5.15: If matrices A and B commute, then A must commute with BT as well.
 5.16: If A is any matrix with ker(A) = {0}, then the matrix AAT represent...
 5.17: If A and B are symmetric n n matrices, then ABBA must be symmetric ...
 5.18: If matrices A and B commute, then matrices AT and BT must commute a...
 5.19: There exists a subspace V of R5 such that dim(V) = dim(V ), where V...
 5.20: Every invertible matrix A can be expressed as the product of an ort...
 5.21: The determinant of all orthogonal 2 2 matrices is 1.
 5.22: If A is any square matrix, then matrix 1 2 (A AT ) is skewsymmetric.
 5.23: The entries of an orthogonal matrix are all less than or equal to 1.
 5.24: Every nonzero subspace of Rn has an orthonormal basis.
 5.25: 3 4 4 3 is an orthogonal matrix.
 5.26: If V is a subspace of Rn and x is a vector in Rn, then vector projV...
 5.27: If A and B are orthogonal 2 2 matrices, then AB = B A.
 5.28: If A is a symmetric matrix, vector v is in the image of A, and w is...
 5.29: The formula ker(A) = ker(AT A) holds for all matrices A
 5.30: If AT A = AAT for an n n matrix A, then A must be orthogonal.
 5.31: There exist orthogonal 22 matrices A and B such that A + B is ortho...
 5.32: If Axx for all x in Rn, then A must represent the orthogonal projec...
 5.33: If A is an invertible matrix such that A1 = A, then A must be ortho...
 5.34: If the entries of two vectors v and w in Rn are all positive, then ...
 5.35: The formula (ker B) = im(BT ) holds for all matrices B.
 5.36: The matrix AT A is symmetric for all matrices A.
 5.37: If matrix A is similar to B and A is orthogonal, then B must be ort...
 5.38: The formula im(B) = im(BT B) holds for all square matrices B.
 5.39: If matrix A is symmetric and matrix S is orthogonal, then matrix S1...
 5.40: If A is a square matrix such that AT A = AAT , then ker(A) = ker(AT ).
 5.41: Any square matrix can be written as the sum of a symmetric and a sk...
 5.42: . If x1, x2,..., xn are any real numbers, then the inequality 9 n k...
 5.43: If AAT = A2 for a 2 2 matrix A, then A must be symmetric.
 5.44: If V is a subspace of Rn and x is a vector in Rn, then the inequali...
 5.45: If A is an n n matrix such that Au = 1 for all unit vectors u, then...
 5.46: If A is any symmetric 2 2 matrix, then there must exist a real numb...
 5.47: There exists a basis of R22 that consists of orthogonal matrices.
 5.48: f A = 1 2 2 1 , then the matrix Q in the Q R factorization of A is ...
 5.49: There exists a linear transformation L from R33 to R22 whose kernel...
 5.50: If a 3 3 matrix A represents the orthogonal projection onto a plane...
Solutions for Chapter 5: Linear Algebra with Applications 5th Edition
Full solutions for Linear Algebra with Applications  5th Edition
ISBN: 9780321796974
Solutions for Chapter 5
Get Full SolutionsChapter 5 includes 50 full stepbystep solutions. Linear Algebra with Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780321796974. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 5. Since 50 problems in chapter 5 have been answered, more than 1085 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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

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.

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.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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

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

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.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

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.

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