 2.1: TRUE OR FALSE? If A is any invertible n x n matrix, then rref(A) = In.
 2.2: TRUE OR FALSE? The formula (A2)1 = (A1)2 holds for all invertible...
 2.3: TRUE OR FALSE? The formula AB = BA holds for all n x n matrices A a...
 2.4: TRUE OR FALSE? If AB = In for two n x n matrices A and B, then A mu...
 2.5: TRUE OR FALSE? If A is a 3 x 4 matrix and B is a 4 x 5 matrix, then...
 2.6: TRUE OR FALSE? The function TX y _y_ l is a linear transformation.
 2.7: TRUE OR FALSE? The matrix represents a rotation combinecwith a scal...
 2.8: TRUE OR FALSE? If A is any invertible n x n matrix, then A commute ...
 2.9: TRUE OR FALSE? The function T formation.
 2.10: TRUE OR FALSE? Matrix1/2  1/2 1/2 1/2represents a rotation.
 2.11: TRUE OR FALSE? There exists a real number k such that the matrix ' ...
 2.12: TRUE OR FALSE? Matrix 0.6 0.8  0.8  0.6represents a rotation.
 2.13: TRUE OR FALSE? The formula det(2A) = 2det(A) holds for all 2 x 2 ma...
 2.14: TRUE OR FALSE? There exists a matrix A such that 1 2 '5 6 1 r 3 4 A...
 2.15: TRUE OR FALSE? Matrix 1 2 3 6is invertible.
 2.16: TRUE OR FALSE? Matrix 1 1 1 1 0 1 1 1 0is invertible
 2.17: TRUE OR FALSE? There exists an upper triangular 2 x2 matrix A such ...
 2.18: TRUE OR FALSE? The function TX '(y + 1)2 (>> l ) 2' y_ _(x 3)2 (x...
 2.19: TRUE OR FALSE? Matrixis invertible for all real numbers k.
 2.20: TRUE OR FALSE? There exists a real number k such that the matrix '*...
 2.21: TRUE OR FALSE? The matrix product scalar multiple of I2.
 2.22: TRUE OR FALSE? There exists a nonzero upper triangular 2 x 2 matrix...
 2.23: TRUE OR FALSE? There exists a positive integer n such that 0  ll" ...
 2.24: TRUE OR FALSE? There exists an invertible 2 x 2 matrix A such that ...
 2.25: TRUE OR FALSE? There exists an invertible nxn matrix with two ident...
 2.26: TRUE OR FALSE? ^ A2 = //j, then matrix A must be invertible.
 2.27: TRUE OR FALSE? There exists a matrix A such that A
 2.28: TRUE OR FALSE? There exists a matrix A such that
 2.29: TRUE OR FALSE? The matrix 1 1 1 1represents a reflection about a l...
 2.30: TRUE OR FALSE? for all real numbers k.
 2.31: TRUE OR FALSE? f matrix d e f is invertible, then matrixa 0 d e8 hm...
 2.32: TRUE OR FALSE?If A2 is invertible, then matrix A itself must be inv...
 2.33: TRUE OR FALSE? If A17 = /2, then matrix A must be I2.
 2.34: TRUE OR FALSE? If A2 = 12, then matrix A must be either I2 or 12
 2.35: TRUE OR FALSE? If matrix A is invertible, then matrix 5 A must be i...
 2.36: TRUE OR FALSE? If A and B are two 4 x 3 matrices such that Av = Bv ...
 2.37: TRUE OR FALSE? If matrices A and B commute, then the formula A2B = ...
 2.38: TRUE OR FALSE? If A2 = A for an invertible nxn matrix A, then A mus...
 2.39: TRUE OR FALSE? If matrices A and B are both invertible, then matrix...
 2.40: TRUE OR FALSE? The equation A2 = A holds for all 2 x 2 matrices A r...
 2.41: TRUE OR FALSE? The equation A1 = A holds for all 2 x 2 matrices A ...
 2.42: TRUE OR FALSE? The formula (Au) (Aw) = v w holds for all invertible...
 2.43: TRUE OR FALSE? There exist a 2 x 3 matrix A and a 3 x 2 matrix B su...
 2.44: TRUE OR FALSE? There exist a 3 x 2 matrix A and a 2 x 3 matrix B su...
 2.45: TRUE OR FALSE? If A2 + 3A + 4/3 = 0 for a 3 x 3 matrix A, then A mu...
 2.46: TRUE OR FALSE? If A is an n x matrix such that A2 = 0, then matrix ...
 2.47: TRUE OR FALSE? If matrix A commutes with 5, and B commutes with C, ...
 2.48: TRUE OR FALSE? If 7 is any linear transformation from R3 to R3, the...
 2.49: TRUE OR FALSE? There exists an invertible 10x10 matrix that has 92 ...
 2.50: TRUE OR FALSE? The formula rref(Afl) = rref(A) rref(B) holds for al...
 2.51: TRUE OR FALSE? There exists an invertible matrix S such that 0 1 0 ...
 2.52: TRUE OR FALSE? If the linear system A2x = b is consistent, then the...
 2.53: TRUE OR FALSE? There exists an invertible 2 x 2 matrix A such that ...
 2.54: TRUE OR FALSE? There exists an invertible 2 x 2 matrix A such thatA2 =
 2.55: TRUE OR FALSE? If a matrix A = represents the orthogonal projection...
 2.56: TRUE OR FALSE? If A is an invertible 2 x 2 matrix and B is any 2 x ...
Solutions for Chapter 2: Linear Transformations
Full solutions for Linear Algebra with Applications  4th Edition
ISBN: 9780136009269
Solutions for Chapter 2: Linear Transformations
Get Full SolutionsLinear Algebra with Applications was written by and is associated to the ISBN: 9780136009269. This expansive textbook survival guide covers the following chapters and their solutions. Since 56 problems in chapter 2: Linear Transformations have been answered, more than 15448 students have viewed full stepbystep solutions from this chapter. Chapter 2: Linear Transformations includes 56 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 4.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

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

Column space C (A) =
space of all combinations of the columns of A.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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

Iterative method.
A sequence of steps intended to approach the desired solution.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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

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

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

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.

Solvable system Ax = b.
The right side b is in the column space of A.

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