 4.6.1: Which of the fol1owi ng sets oi vectors are bases for 21
 4.6.2: Which of the following sets of vectors are bases for 1?
 4.6.3: Which of the followi ng SCts of vectors are bases for ~?
 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: In) AII,,"o~of''''fo= ~lW O"b~"+' Ih) All ,,"o~ of 'ho fonn [~] . w...
 4.6.20: la) AII''''O~Of'h''O=[~lWh'''O~O [ "+'] a  b (b) All vectors of th...
 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 M
 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 polynomi vectors in S. Prove...
 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: 9780471669593
Solutions for Chapter 4.6: Basis and Dimension
Get Full SolutionsChapter 4.6: Basis and Dimension includes 49 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780471669593. Since 49 problems in chapter 4.6: Basis and Dimension have been answered, more than 9195 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9.

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

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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.

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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.

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 matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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.

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).

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

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

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