 2.8.1E: Exercises display sets in ?2. Assume the sets include the bounding ...
 2.8.2E: Exercises display sets in ?2. Assume the sets include the bounding ...
 2.8.3E: Exercises display sets in ?2. Assume the sets include the bounding ...
 2.8.4E: Exercises display sets in ?2. Assume the sets include the bounding ...
 2.8.5E: Let and . Determine if w is in the subspace of ?3 generated by v1 a...
 2.8.6E: Let and . Determine if u is in the subspace of ?4 generated by {v1,...
 2.8.7E: Let a. How many vectors are in ?b. How many vectors are in Col A?c....
 2.8.8E: Let and . Determine if p is in Col A, where A = [v1 v2v3].
 2.8.9E: With A and p as in Exercise 7, determine if p is in Nul A.Exercise ...
 2.8.10E: With u = (?2, 3, 1) and A as in Exercise, determine if u is in Nul ...
 2.8.11E: In Exercises 11 and 12, give integers p and q such that Nul A is a ...
 2.8.12E: In Exercises, give integers p and q such that Nul A is a subspace o...
 2.8.13E: For A as in Exercise 11, find a nonzero vector in Nul A and a nonze...
 2.8.14E: For A as in Exercise 12, find a nonzero vector in Nul A and a nonze...
 2.8.15E: Determine which sets in Exercises are bases for ?2 or ?3. Justify e...
 2.8.16E: Determine which sets in Exercises are bases for ?2 or ?3. Justify e...
 2.8.17E: Determine which sets in Exercises are bases for ?2 or ?3. Justify e...
 2.8.18E: Determine which sets in Exercises are bases for ?2 or ?3. Justify e...
 2.8.19E: Determine which sets in Exercises 15–20 are bases for or . Justify ...
 2.8.20E: Determine which sets in Exercises are bases for ?2 or ?3. Justify e...
 2.8.21E: In Exercises 21 and 22, mark each statement True or False. Justify ...
 2.8.22E: In Exercises, mark each statement True or False. Justify each answe...
 2.8.23E: Exercises 23–26 display a matrix A and an echelon form of A. Find a...
 2.8.24E: Exercises display a matrix A and an echelon form of A. Find a basis...
 2.8.25E: Exercises 23–26 display a matrix A and an echelon form of A. Find a...
 2.8.26E: Exercises display a matrix A and an echelon form of A. Find a basis...
 2.8.27E: Construct a 3 × 3 matrix A and a nonzero vector b such that b is in...
 2.8.28E: Construct a 3 × 3 matrix A and a vector b such that b is not in Col A.
 2.8.29E: Construct a nonzero 3 × 3 matrix A and a nonzero vector b such that...
 2.8.30E: Suppose the columns of a matrix are linearly independent. Explain w...
 2.8.31E: In Exercises 31–36, respond as comprehensively as possible, and jus...
 2.8.32E: In Exercises, respond as comprehensively as possible, and justify y...
 2.8.33E: In Exercises, respond as comprehensively as possible, and justify y...
 2.8.34E: In Exercises, respond as comprehensively as possible, and justify y...
 2.8.35E: In Exercises, respond as comprehensively as possible, and justify y...
 2.8.36E: In Exercises, respond as comprehensively as possible, and justify y...
 2.8.37E: [M] In Exercises 37 and 38, construct bases for the column space an...
 2.8.38E: [M] In Exercises, construct bases for the column space and the null...
Solutions for Chapter 2.8: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 2.8
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 2.8 includes 38 full stepbystep solutions. Since 38 problems in chapter 2.8 have been answered, more than 43832 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384.

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).

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.

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.

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

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

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

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).

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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).

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

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

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

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).