 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 Sieva Kozinsky and is associated to the ISBN: 9780321796974. Since 67 problems in chapter 4 have been answered, more than 1093 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 5.

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).

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.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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 .

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

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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·
I don't want to reset my password
Need help? Contact support
Having trouble accessing your account? Let us help you, contact support at +1(510) 9441054 or support@studysoup.com
Forgot password? Reset it here