 4.1: The polynomials of degree less than 7 form a sevendimensional subsp...
 4.2: The function T ( f ) = 3 f 4 f from C to C is a linear transformation.
 4.3: . The lower triangular 2 2 matrices form a subspace of the space of...
 4.4: The kernel of a linear transformation is a subspace of the domain.
 4.5: . The space R23 is fivedimensional.
 4.6: If f1,..., fn is a basis of a linear space V, then any element of V...
 4.7: The space P1 is isomorphic to C.
 4.8: If the kernel of a linear transformation T from P4 to P4 is {0}, th...
 4.9: If W1 and W2 are subspaces of a linear space V, then the intersecti...
 4.10: If T is a linear transformation from P6 to R22, then the kernel of ...
 4.11: All bases of P3 contain at least one polynomial of degree 2.
 4.12: If T is an isomorphism, then T 1 must be an isomorphism as well.
 4.13: The linear transformation T ( f ) = f + f from C to C is an isomorp...
 4.14: All linear transformations from P3 to R22 are isomorphisms.
 4.15: If T is a linear transformation from V to V, then the intersection ...
 4.16: The space of all upper triangular 4 4 matrices is isomorphic to the...
 4.17: Every polynomial of degree 3 can be expressed as a linear combinati...
 4.18: If a linear space V can be spanned by 10 elements, then the dimensi...
 4.19: The function T (M) = det(M) from R22 to R is a linear transformation.
 4.20: There exists a 2 2 matrix A such that the space V of all matrices c...
 4.21: The linear transformation T (M) = 1 2 3 6 M from R22 to R22 has ran...
 4.22: If the matrix of a linear transformation T (with respect to some ba...
 4.23: The kernel of the linear transformation T f (t) = f (t2) from P to ...
 4.24: If S is any invertible 2 2 matrix, then the linear transformation T...
 4.25: There exists a 2 2 matrix A such that the space of all matrices com...
 4.26: There exists a basis of R22 that consists of four invertible matrices.
 4.27: If the image of a linear transformation T from P to P is all of P, ...
 4.28: If f1, f2, f3 is a basis of a linear space V, then f1, f1 + f2, f1 ...
 4.29: If a, b, and c are distinct real numbers, then the polynomials (x b...
 4.30: The linear transformation T f (t) = f (4t 3) from P to P is an isom...
 4.31: If W is a subspace of V, and if W is finite dimensional, then V mus...
 4.32: There exists a linear transformation from R33 to R22 whose kernel c...
 4.33: Every twodimensional subspace of R22 contains at least one inverti...
 4.34: If = ( f, g) and = ( f, f + g) are two bases of a linear space V, t...
 4.35: . If the matrix of a linear transformation T with respect to a basi...
 4.36: The linear transformation T ( f ) = f from Pn to Pn has rank n, for...
 4.37: If the matrix of a linear transformation T (with respect to some ba...
 4.38: There exists a subspace of R34 that is isomorphic to P9.
 4.39: There exist two distinct subspaces W1 and W2 of R22 whose union W1 ...
 4.40: There exists a linear transformation from P to P5 whose image is al...
 4.41: If f1,..., fn are polynomials such that the degree of fk is k (for ...
 4.42: The transformation D( f ) = f from C to C is an isomorphism.
 4.43: If T is a linear transformation from P4 to W with im(T ) = W, then ...
 4.44: The kernel of the linear transformation T f (t) = . 1 0 f (t) dt fr...
 4.45: If T is a linear transformation from V to V, then { f in V : T ( f ...
 4.46: If T is a linear transformation from P6 to P6 that transforms t k i...
 4.47: There exist invertible 2 2 matrices P and Q such that the linear tr...
 4.48: There exists a linear transformation from P6 to C whose kernel is i...
 4.49: If f1, f2, f3 is a basis of a linear space V, and if f is any eleme...
 4.50: There exists a twodimensional subspace of R22 whose nonzero elemen...
 4.51: The space P11 is isomorphic to R34 .
 4.52: If T is a linear transformation from V to W, and if both im(T ) and...
 4.53: If T is a linear transformation from V to R22 with ker(T ) = {0}, t...
 4.54: The function T f (t) = d dt . 3t+4 2 f
 4.55: Any fourdimensional linear space has infinitely many threedimensi...
 4.56: If the matrix of a linear transformation T (with respect to some ba...
 4.57: If the image of a linear transformation T is infinite dimensional, ...
 4.58: There exists a 2 2 matrix A such that the space of all matrices com...
 4.59: If A, B, C, and D are noninvertible 2 2 matrices, then the matrices...
 4.60: There exist two distinct threedimensional subspaces W1 and W2 of P...
 4.61: If the elements f1,..., fn (where f1 = 0) are linearly dependent, t...
 4.62: There exists a 3 3 matrix P such that the linear transformation T (...
 4.63: If f1, f2, f3, f4, f5 are elements of a linear space V, and if ther...
 4.64: There exists a linear transformation T from P6 to P6 such that the ...
 4.65: If T is a linear transformation from V to W, and if both im(T ) and...
 4.66: If the matrix of a linear transformation T (with respect to some ba...
 4.67: Every threedimensional subspace of R22 contains at least one inver...
Solutions for Chapter 4: Linear Algebra with Applications 5th Edition
Full solutions for Linear Algebra with Applications  5th Edition
ISBN: 9780321796974
Solutions for Chapter 4
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 4 includes 67 full stepbystep solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780321796974. Since 67 problems in chapter 4 have been answered, more than 2263 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 5.

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

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

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

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).

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.

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).

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

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

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

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

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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

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·

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