 4.9.1: Find a basis for the subspace V of R3 splmned by S~ ! [i]. m [=;]. ...
 4.9.2: Find a basis for the subspace of P3 spanned by S = {13 +1 2 +2t + L...
 4.9.3: Find a basis for the subspace of M22 spanned by s~ l[ : ;][; :][~ ;...
 4.9.4: Find a basis for the subspace of R2 spanned by S~I[I 2].[2 3].[3 1]...
 4.9.5: /11 Exercises 5 and 6, find a basi.l" for the rolV .Ipace of A COII...
 4.9.6: /11 Exercises 5 and 6, find a basi.l" for the rolV .Ipace of A COII...
 4.9.7: III Exercises 7 alld 8. filld a basi.I' for the CO/1111111 JjX1Ce o...
 4.9.8: III Exercises 7 alld 8. filld a basi.I' for the CO/1111111 JjX1Ce o...
 4.9.9: III Ext'lrises 9 alld 10. filld the row and colllllln rallb of Ihe ...
 4.9.10: III Ext'lrises 9 alld 10. filld the row and colllllln rallb of Ihe ...
 4.9.11: Let A be an III x n malrix in row echelon form. Prove that rank A =...
 4.9.12: For each of the following matrices. verify Theorem 4.18 by computin...
 4.9.13: III EI"{'rciJes /J alld 14, compute Ihe milk alld IIl1l1ity oj {'ac...
 4.9.14: III EI"{'rciJes /J alld 14, compute Ihe milk alld IIl1l1ity oj {'ac...
 4.9.15: Which of the following matrices are equivalent? A~ B~ c= U [t [ ~ [...
 4.9.16: III Erercisl's 16 and 17. delennint which of the gil'l'lIlinear sys...
 4.9.17: III Erercisl's 16 and 17. delennint which of the gil'l'lIlinear sys...
 4.9.18: III Exercises 18 and 19. lise CO/vllw)' 4.7 /() find which oflhe gi...
 4.9.19: III Exercises 18 and 19. lise CO/vllw)' 4.7 /() find which oflhe gi...
 4.9.20: III t."xerciJ/'s 20 and 21. lI~e Corollary 4.8 to find Ollt lI"ilnh...
 4.9.21: III t."xerciJ/'s 20 and 21. lI~e Corollary 4.8 to find Ollt lI"ilnh...
 4.9.22: III rercis/'s 22 alld 23. lise Corollary 4.9 10 find which of the g...
 4.9.23: III rercis/'s 22 alld 23. lise Corollary 4.9 10 find which of the g...
 4.9.24: III Tercil"e.\' 24 alld 25, filld milk .'\, by obtaillillg (/ matri...
 4.9.25: III Tercil"e.\' 24 alld 25, filld milk .'\, by obtaillillg (/ matri...
 4.9.26: III El"erci.I'es 26 alld 27. lise Corolllll)' 4.10 10 determine whe...
 4.9.27: III El"erci.I'es 26 alld 27. lise Corolllll)' 4.10 10 determine whe...
 4.9.28: In Excercise 28 through 33. soh'e using Ihe cOllcepl of mnk.
 4.9.29: In Excercise 28 through 33. soh'e using Ihe cOllcepl of mnk.
 4.9.30: In Excercise 28 through 33. soh'e using Ihe cOllcepl of mnk.
 4.9.31: In Excercise 28 through 33. soh'e using Ihe cOllcepl of mnk.
 4.9.32: In Excercise 28 through 33. soh'e using Ihe cOllcepl of mnk.
 4.9.33: In Excercise 28 through 33. soh'e using Ihe cOllcepl of mnk.
 4.9.34: (a) If A is a 3 x 4 matrix. what is the largest possible value for ...
 4.9.35: Let A be a 7 x 3 matrix whose mnk is 3. (a) Are the rows of A linea...
 4.9.36: Let A be a 3 x 5 matrix. (a) Give a/l possible values for the rank ...
 4.9.37: Let S = [v ,. V2 .. vn) be a set of II vectors in R" and let A be t...
 4.9.38: Let S= (v, . I'! ..... v.lbe a setof" vectOis in R',and let A be th...
 4.9.39: Let A be an n x II matrix. Show that the homogeneous system Ax = 0 ...
 4.9.40: Let A be an II x n matrix. Show that rank A = II if and only if the...
 4.9.41: Let A be an II x /I matrix. Prove that the rows of A are linearly i...
 4.9.42: Let S = (VI. V2 ... , vd be a basis for a subspace V of R" that is ...
 4.9.43: Let S = {VI. V2 v.) be an ordered basis for 3J1 11  dimensional vc...
 4.9.44: Let A be an III x II matrix. SllOw that the linear system Ax = b ha...
 4.9.45: Let A be an III x II matrix with //I 1= II. Show that either lhe ro...
 4.9.46: Suppose that the linear system Ax = h, where A is III XII. IS consi...
 4.9.47: What can you say about the dimension of the solution 'pal:e uf a hu...
 4.9.48: What can you say about the dimension of the solution 'pal:e uf a hu...
 4.9.49: Show that a set S = ( VI . V2 ... v") of vectors in R" ( R.) spans ...
 4.9.50: Determine whether your software has a commaoo for computing the ran...
 4.9.51: Determine whether your software has a commaoo for computing the ran...
Solutions for Chapter 4.9: Coordinates and Isomorphisms
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780471669593
Solutions for Chapter 4.9: Coordinates and Isomorphisms
Get Full SolutionsSince 51 problems in chapter 4.9: Coordinates and Isomorphisms have been answered, more than 10247 students have viewed full stepbystep solutions from this chapter. Chapter 4.9: Coordinates and Isomorphisms includes 51 full stepbystep solutions. This 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. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780471669593.

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

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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

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.

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

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.

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.

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.

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

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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

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!

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

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

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

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).