 6.4.7: (Calcllllls Required) Let Ll PI + P2 be the linear transformation ...
 6.4.8: Let LI. L,. and S be as in Exercise 3. Find the following: (a) (L 1...
 6.4.9: If [! _: iJ , .. he "pre~m"'o<' of' h'"'' 0",' ator L: R 3 + R3 w...
 6.4.10: Let L j L,. and L3 be linear transformations of R3 into R, defined ...
 6.4.11: Find the dimension of the vector space U of all linear transformati...
 6.4.12: Repeat Exercise II for each of the fo llowing: (a ) V = IV is the v...
 6.4.13: Let A = [ali ] be a given /II x I! matrix. and let V and IV be give...
 6.4.14: Let A = [~ ~ ~ 1 Let S be the natural basis for Rl and T be the nat...
 6.4.15: Let A be as in Exercise 14. Consider the ordered bases S = if'. I. ...
 6.4.16: Find two linear transformations L j : H' 4 H: and L 2: H2 _ R'such...
 6.4.17: Find a linear transformation L : R2 4 K'. L '" f, the identity ope...
 6.4.18: Find a linear transformation L: H' ........ H'. L '" O. the zero tr...
 6.4.19: Find a linear transformation L: R 2 ) H2. L I I. L '" O ,such th...
 6.4.20: Let L : R 3 ) Rl be the linear transformation defined in Exercise...
 6.4.21: Let L : R 3 ..... R3 be the linear transformation definoo in Exerci...
 6.4.22: Let L : R' ..... Rl be the invertible linear transfonr.ation repres...
 6.4.23: Let L: V ..... V be a linear transfonnation represented by a matrix...
 6.4.24: Let L: P, + P, be the invertible linear transfonr.ation represente...
Solutions for Chapter 6.4: Vector Space of Matrices and Vector Space of Linear Transformations (Optional)
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780132296540
Solutions for Chapter 6.4: Vector Space of Matrices and Vector Space of Linear Transformations (Optional)
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. Since 18 problems in chapter 6.4: Vector Space of Matrices and Vector Space of Linear Transformations (Optional) have been answered, more than 12141 students have viewed full stepbystep solutions from this chapter. Chapter 6.4: Vector Space of Matrices and Vector Space of Linear Transformations (Optional) includes 18 full stepbystep solutions. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780132296540.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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!).

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.

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.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.

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.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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).

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

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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