 5.1: TRUE OR FALSE? If A and B are symmetric nxn matrices, then A + B mu...
 5.2: TRUE OR FALSE? If matrices A and S are orthogonal, then S~[ AS is o...
 5.3: TRUE OR FALSE? All nonzero symmetric matrices are invertible.
 5.4: TRUE OR FALSE? If A is an n x n matrix such that AAT = then A must ...
 5.5: TRUE OR FALSE? If u is a unit vector in R, and L = span(iJ), then p...
 5.6: TRUE OR FALSE? If A is a symmetric matrix, then 1A must be symmetri...
 5.7: TRUE OR FALSE? If T is a linear transformation from R" to R" such t...
 5.8: TRUE OR FALSE? If A is an invertible matrix, then the equation (AT)...
 5.9: TRUE OR FALSE? If matrix A is orthogonal, then matrix A2 must be or...
 5.10: TRUE OR FALSE? The equation (AB)T = Ar BT holds for all n x n matri...
 5.11: TRUE OR FALSE? If matrix A is orthogonal, then AT must be orthogona...
 5.12: TRUE OR FALSE? . If A and B are symmetric nxn matrices, then A B mu...
 5.13: TRUE OR FALSE? If matrices A and B commute, then A must commute wit...
 5.14: TRUE OR FALSE? If ,4 is any matrix with ker(A) = {0}, then the matr...
 5.15: TRUE OR FALSE? If A and B are symmetric n x n matrices, then ABBA m...
 5.16: TRUE OR FALSE? If matrices A and B commute, then matrices AT and BT...
 5.17: TRUE OR FALSE? There exists a subspace V of R5 such that dim(V) = d...
 5.18: TRUE OR FALSE? Every invertible matrix A can be expressed as the pr...
 5.19: TRUE OR FALSE? If jc and v are two vectors in R", then the equation...
 5.20: TRUE OR FALSE? The equation det(A7 ) = det(A) holds for all 2 x 2 m...
 5.21: TRUE OR FALSE? If A and B are orthogonal 2x2 matrices, then AB = BA.
 5.22: TRUE OR FALSE? If A is a symmetric matrix, vector 5 is in the image...
 5.23: TRUE OR FALSE? The formula ker(A) = ker {A7 A) holds for all matric...
 5.24: TRUE OR FALSE? If A7 A = AAT for an n x n matrix A, then A must be ...
 5.25: TRUE OR FALSE? The determinant of all orthogonal 2 x 2 matrices is 1.
 5.26: TRUE OR FALSE? If A is any square matrix, then matrix (A A7) is sk...
 5.27: TRUE OR FALSE? The entries of an orthogonal matrix are all less tha...
 5.28: TRUE OR FALSE? Every nonzero subspace of R has an orthonormal basis,
 5.29: TRUE OR FALSE? is an orthogonal matrix.
 5.30: TRUE OR FALSE? If V is a subspace of R/? and Jc is a vector in R '\...
 5.31: TRUE OR FALSE? There exist orthogonal 2x2 matrices A and B such tha...
 5.32: TRUE OR FALSE? If  AJc < Jf for all Jc in R '\ then A must ...
 5.33: TRUE OR FALSE? If A is an invertible matrix such that A1 = A, then...
 5.34: TRUE OR FALSE? If the entries of two vectors 5 and w in R" are all ...
 5.35: TRUE OR FALSE? The formula (ker# )1 = im(BT) holds for all matrices B.
 5.36: TRUE OR FALSE? The matrix AT A is symmetric for all matrices A.
 5.37: TRUE OR FALSE? If matrix A is similar to B and A is orthogonal, the...
 5.38: TRUE OR FALSE? The formula \m(B) = im(BTB) holds for ail square mat...
 5.39: TRUE OR FALSE? If matrix A is symmetric and matrix S is orthogonal,...
 5.40: TRUE OR FALSE? If A is a square matrix such that AT A = AAT, then k...
 5.41: TRUE OR FALSE? Any square matrix can be written as the sum of a sym...
 5.42: TRUE OR FALSE? If x\, X 2....... xn are any real numbers, then the ...
 5.43: TRUE OR FALSE? If AAt = A2 for a 2 x 2 matrix A, then A must be sym...
 5.44: TRUE OR FALSE? If V is a subspace of IR" and x is a vector in IR", ...
 5.45: TRUE OR FALSE? If A is an n x n matrix such that  Au\\ = 1 for al...
 5.46: TRUE OR FALSE? If A is any symmetric 2x2 matrix, then there must ex...
 5.47: TRUE OR FALSE? There exists a basis of R2x2 that consists of orthog...
 5.48: TRUE OR FALSE? If A = , then the matrix Q in the QR factoriza2 1 ti...
 5.49: TRUE OR FALSE? There exists a linear transformation L from R3x3 to ...
 5.50: TRUE OR FALSE? If a 3 x 3 matrix A represents the orthogonal projec...
Solutions for Chapter 5: Orthogonality and Least Squares
Full solutions for Linear Algebra with Applications  4th Edition
ISBN: 9780136009269
Solutions for Chapter 5: Orthogonality and Least Squares
Get Full SolutionsLinear Algebra with Applications was written by and is associated to the ISBN: 9780136009269. Since 50 problems in chapter 5: Orthogonality and Least Squares have been answered, more than 46282 students have viewed full stepbystep solutions from this chapter. Chapter 5: Orthogonality and Least Squares includes 50 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 4. This expansive textbook survival guide covers the following chapters and their solutions.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Block matrix.
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.

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

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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).

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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).

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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 ().

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.