 4.3.24E: Let be a linearly independent set in Rn. Explain why B must be a ba...
 4.3.1E: Determine whether the sets in Exercises 1–8 are bases for . Of the ...
 4.3.2E: Determine which sets in are bases for ?3. Of the sets that are not ...
 4.3.3E: Determine which sets in are bases for ?3. Of the sets that are not ...
 4.3.4E: Determine which sets in are bases for ?3. Of the sets that are not ...
 4.3.5E: Determine which sets in are bases for ?3. Of the sets that are not ...
 4.3.6E: Determine which sets in are bases for ?3. Of the sets that are not ...
 4.3.7E: Determine whether the sets in Exercises 1–8 are bases for . Of the ...
 4.3.8E: Determine which sets in are bases for ?3. Of the sets that are not ...
 4.3.9E: Find bases for the null spaces of the matrices given in. Refer to t...
 4.3.10E: Find bases for the null spaces of the matrices given in. Refer to t...
 4.3.11E: Find a basis for the set of vectors in ?3 in the plane × + 2y + z =...
 4.3.12E: Find a basis for the set of vectors in ?2 on the line y = 5x.
 4.3.13E: In Exercises 13 and 14, assume that A is row equivalent to B. Find ...
 4.3.14E: In assume that A is row equivalent to B. Find bases for Nul A and C...
 4.3.15E: Find a basis for the space spanned by the given vectors, v1,..., v5.
 4.3.16E: Find a basis for the space spanned by the given vectors, v1,..., v5.
 4.3.17E: Find a basis for the space spanned by the given vectors, v1,..., v5.
 4.3.18E: Find a basis for the space spanned by the given vectors, v1,..., v5.
 4.3.19E: and also let It can be verified that Use this information to find a...
 4.3.20E: Let It can be verified that v1 3v2 + 5v3 = 0. Use this information...
 4.3.21E: In Exercises 21 and 22, mark each statement True or False. Justify ...
 4.3.22E: In Exercises 21 and 22, mark each statement True or False. Justify ...
 4.3.23E: Suppose Is a basis for
 4.3.25E: and let H be the set of vectors in whose second and third entries a...
 4.3.26E: In the vector space of all realvalued functions, find a basis for ...
 4.3.27E: Let V be the vector space of functions that describe the vibration ...
 4.3.28E: (RLC circuit) The circuit in the figure consists of a resistor (R o...
 4.3.29E: Exercises 29 and 30 show that every basis for must contain exactly ...
 4.3.30E: Exercises 29 and 30 show that every basis for must contain exactly ...
 4.3.31E: Show that if is linearly dependent in V, then the set of images, , ...
 4.3.32E: Suppose that T is a onetoone transformation, so that an equation ...
 4.3.33E: Consider the polynomials a linearly independent set in ? Why or why...
 4.3.34E: Consider the polynomials By inspection, write a linear dependence r...
 4.3.35E: Let V be a vector space that contains a linearly independent set De...
 4.3.36E: Find bases for H, K, and H C K. (See Exercises 33 and 34 in Section...
 4.3.37E: [M] Show that is a linearly independent set of functions defined on...
 4.3.38E: [M] Show that is a linearly independentset of functions defined on ...
Solutions for Chapter 4.3: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 4.3
Get Full SolutionsLinear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. Since 38 problems in chapter 4.3 have been answered, more than 43682 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5. Chapter 4.3 includes 38 full stepbystep solutions.

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

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.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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.

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.

Outer product uv T
= column times row = rank one matrix.

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).

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

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

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

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

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.

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