 5.5.1: Let W be the subspace of R.1 spanned by the vector (3) Find a basis...
 5.5.2: Let IV ~ 'P'" I [J . [n I (3) Find a basis for W.i. (b ) Describe ...
 5.5.3: Let \V be the subspace of Rs spanned by the vectors 11',. W 1. 11...
 5.5.4: Let \V be the subspace of R4 spanned by the vectors 11'1. W 1. 11...
 5.5.5: Let \V be the subspace of R4 spanned by the vectors 11'1. W 1. 11...
 5.5.6: Let \V be the subspace of R4 spanned by the vectors 11'1. W 1. 11...
 5.5.7: Let \V be the subspace of R4 spanned by the vectors 11'1. W 1. 11...
 5.5.8: Let \V be the subspace of R4 spanned by the vectors 11'1. W 1. 11...
 5.5.9: III Exaci!le!l 9 alld 10, cOlllpllfe Ihe jOllr fillldwlIl'lila/ll'c...
 5.5.10: III Exaci!le!l 9 alld 10, cOlllpllfe Ihe jOllr fillldwlIl'lila/ll'c...
 5.5.11: III ExerciJe.l II throllgh /4,jilld projwv jor Ihe gil'ell I'ector ...
 5.5.12: III ExerciJe.l II throllgh /4,jilld projwv jor Ihe gil'ell I'ector ...
 5.5.13: III ExerciJe.l II throllgh /4,jilld projwv jor Ihe gil'ell I'ector ...
 5.5.14: III ExerciJe.l II throllgh /4,jilld projwv jor Ihe gil'ell I'ector ...
 5.5.15: Let IV be the subspncc of Rl wilh ol1hononnnl basis [WI. W2\, where...
 5.5.16: Let \V be the subspace of R4 with otthonormal basis [WI. ',1,'2. W3...
 5.5.17: Let IV be the plane in R3 given by the equation x  )"  , ~ o. W,;...
 5.5.18: Let IV be the plane in R3 given by the equation x  )"  , ~ o. W,;...
 5.5.19: Let IV be the subspace of Rl defined in Exercise 15. and ,,, , ~ [ ...
 5.5.20: Let IV be the subspace of R4 defined in Exercise 16. and ,,, ' ~ [ ...
 5.5.21: Let IV be the subspace of continuous functions on [ J"[. It[ defin...
 5.5.22: III Exercise.l 22 {llId 23, find the f"tJllrier polynomial of degre...
 5.5.23: III Exercise.l 22 {llId 23, find the f"tJllrier polynomial of degre...
 5.5.24: Show that if V is an itmer product space and IV is II sub space of ...
 5.5.25: Let V be an inner product space. Show that the otthog. onal complem...
 5.5.26: Show Ihal if IV is a Hlbspace of an inner product . pace V that is ...
 5.5.27: Let A be an //I x II matrix. Show that every vector \' in R" can be...
 5.5.28: Let V be a Euclidean space, and IV a sub.>pace of V. Show that if W...
 5.5.29: Let IV be a subspace of an inner product space V and let {WI. w, .....
Solutions for Chapter 5.5: Orthogonal Complements
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780471669593
Solutions for Chapter 5.5: Orthogonal Complements
Get Full SolutionsElementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780471669593. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. Since 29 problems in chapter 5.5: Orthogonal Complements have been answered, more than 9126 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 5.5: Orthogonal Complements includes 29 full stepbystep solutions.

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

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).

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

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.

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.

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

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.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

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.