 4.6.1: Which of the fol1owi ng sets oi vectors are bases for 21 (.j H:][:...
 4.6.2: Which of the following sets of vectors are bases for 1? (. j mlU]) ...
 4.6.3: Which of the followi ng SCts of vectors are bases for ~? (.j 1[1 0 ...
 4.6.4: Which of the following sets of vectors are bases for Pl? (a) ( / 2...
 4.6.5: Which of the fol1owing sets of vectors are bases for p)? (a) (I l +...
 4.6.6: Show that the set of matrices fonns a ba.~is for thc vector space Mn.
 4.6.7: hr t .xercise.\" 7 (///(I 8. ,Ielenllllle It'/rrclr vf tire gll'ell...
 4.6.8: hr t .xercise.\" 7 (///(I 8. ,Ielenllllle It'/rrclr vf tire gll'ell...
 4.6.9: III .rerci.l"el 9 ",,,110. delermille whiclr of lire gil'l'n sl,"se...
 4.6.10: III .rerci.l"el 9 ",,,110. delermille whiclr of lire gil'l'n sl,"se...
 4.6.11: Find a basis for thc subspace IV of Rl spanned by What is the dimen...
 4.6.12: Find 3 ba.~is for the subspace W of ~ spanned by the set of \"cctor...
 4.6.13: Let IV be the subspace of P1 spanned by Itl + t 2 _ 21 + I. t2 + 1....
 4.6.14: Let Find a basis fOf the subspace IV = span S of M 21
 4.6.15: Find 311 values of a for which {[u' 0 iJ.[O u 2].[1 0 III tS a b3Si...
 4.6.16: Find a basis for the subspace IV of M lJ consisting of all symmetri...
 4.6.17: Find 3 basis for the subspace of M ]] consisting o f all diagonal m...
 4.6.18: Let IV be the subspace of the sp"ce of all continuous real valued f...
 4.6.19: 111 rercil'el' /9 ami 20. find a /){lsiJjor Ih t! gil'ell slIbspac...
 4.6.20: 111 rercil'el' /9 ami 20. find a /){lsiJjor Ih t! gil'ell slIbspac...
 4.6.21: Find a basis for the subspace of "2 consisting of all vee tors of t...
 4.6.22: Find a b3Sis for the subspace of p] consisting of all vec tors of t...
 4.6.23: In rercises 23 Will 24, find lite dimell.fiOlls of lite gil'en Sllb...
 4.6.24: In rercises 23 Will 24, find lite dimell.fiOlls of lite gil'en Sllb...
 4.6.25: Find the dimensions of the subSp3Ces of Rl spanned by the vectors i...
 4.6.26: Find the dimensions of Ihe subspnces of RJ spanned by the vectors i...
 4.6.27: Find Ihe dimensions of the bspace~ of /(4 spanned by the vectors in...
 4.6.28: Find a basis for R ] lhal includes I.) lhe ,,"m m (h) Iho ,,"o~ m Md m
 4.6.29: Find a basis for PJ that includes lhe vectors I] + I and ~  I.
 4.6.30: Find a basis for AllJ. What is the dimension of M n? Generalize to ...
 4.6.31: Find the dimension c f the subspace of P! consisling of all ,'cctOT...
 4.6.32: Find the dimension of Ihe subspace of PJ consisling of all vectors ...
 4.6.33: Give an example of a twodimensional subspace of R4.
 4.6.34: Give an example of a twodimensional subspace of 1'3
 4.6.35: Prove that if (v\. V2 .... Vk ) i, a basis for a vector space V. th...
 4.6.36: Prove that the vector space I' of all polynomials is not finitedi...
 4.6.37: Let V be an IIdimensional vector space. Show thal any /I + I vecto...
 4.6.38: Prove Corollary 4.3.
 4.6.39: Prove Corollary 4.4.
 4.6.40: Prove Corollary 4.5
 4.6.41: Show that if IV is a sllbspace ofa finitedimensional veclor space ...
 4.6.42: Show that it" IV is a sllbspace ora tinitedimensional veclor space...
 4.6.43: Prove that the sllbspaces of RJ are (0 ). RJ itself. amI any line o...
 4.6.44: Let S = {VI , \ '2 ... v.] be a set of nonzero vectors in a vector ...
 4.6.45: Prove lbeorem 4.12.
 4.6.46: Prove Theorem 4.13.
 4.6.47: Suppose that (V I. \', .. v. 1 is a basis for R". Show that if A is...
 4.6.48: Suppose Ihat ( VI. V2 ... v") is a linearly independent set of vect...
 4.6.49: Find a basis for the ~ubspace IV of all 3 x 3 matrices with trace e...
Solutions for Chapter 4.6: Basis and Dimension
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780132296540
Solutions for Chapter 4.6: Basis and Dimension
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 4.6: Basis and Dimension includes 49 full stepbystep solutions. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780132296540. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. Since 49 problems in chapter 4.6: Basis and Dimension have been answered, more than 13645 students have viewed full stepbystep solutions from this chapter.

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).

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

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.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

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.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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 .

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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.

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.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

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.

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.

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

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.

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