 5.3.1: Verify that the standard inner product on R" satisfies the properti...
 5.3.2: Verify that the standard inner product on R" satisfies the properti...
 5.3.3: Let V = Mnn be the rcal vector space of all" x" matrices. If A and ...
 5.3.4: Let V = Mnn be the rcal vector space of all" x" matrices. If A and ...
 5.3.5: Let V = Mnn be the rcal vector space of all" x" matrices. If A and ...
 5.3.6: Let V = Mnn be the rcal vector space of all" x" matrices. If A and ...
 5.3.7: Let V be an inner product space. Show the following (a) 11011 = o. ...
 5.3.8: III E.>.erci.le.l 8 and 9. lei V be the Euclideall space R .. with ...
 5.3.9: III E.>.erci.le.l 8 and 9. lei V be the Euclideall space R .. with ...
 5.3.10: III Exerei.le.l 10 alld II. lise rhe inner prvdllcr .Ipace of cOlll...
 5.3.11: III Exerei.le.l 10 alld II. lise rhe inner prvdllcr .Ipace of cOlll...
 5.3.12: III Exercises 12 and 13, let V be Ihe Euclidean s!,ace of Example 3...
 5.3.13: III Exercises 12 and 13, let V be Ihe Euclidean s!,ace of Example 3...
 5.3.14: In /:.'J:ercixes 14 and 15, 11'1 V be the inner plvduel .1'Ix/Ce of...
 5.3.15: In /:.'J:ercixes 14 and 15, 11'1 V be the inner plvduel .1'Ix/Ce of...
 5.3.16: Prove the parallelogram law for any two vectors in an Inner pro
 5.3.17: Let V be an inner product space. Show tbat lI eull clilull for any ...
 5.3.18: State the Caucby Schwarz inequality for the inner prod. uct spaces...
 5.3.19: Let V be an inner product space. Prove that if u and v are any vect...
 5.3.20: Let lu . v. w) be an orthononnal set of vectors in an inner product...
 5.3.21: Let V be an irmer product space. If u and v are vectors In V. show ...
 5.3.22: Let V be the Euclidean space ~ considered in Exercise g. Find which...
 5.3.23: Let V be the Euclidean space ~ considered in Exercise g. Find which...
 5.3.24: For each of the irmer products defined in Examples 3 and 5. choose ...
 5.3.25: If V is an inner product space. prove that the distance function of...
 5.3.26: If V is an inner product space. prove that the distance function of...
 5.3.27: III Exerci.les 27 and 28. Id V be the inlier product SfXice of Emll...
 5.3.28: III Exerci.les 27 and 28. Id V be the inlier product SfXice of Emll...
 5.3.29: III terdse.\" 29 alld 30, which of Ihe gil'en sels of wewrs ill R3....
 5.3.30: III terdse.\" 29 alld 30, which of Ihe gil'en sels of wewrs ill R3....
 5.3.31: III Exerci.l"t,s 31 and 32. Id V be Ihe inner product .IpGce of Exa...
 5.3.32: III Exerci.l"t,s 31 and 32. Id V be Ihe inner product .IpGce of Exa...
 5.3.33: III .11'1Lisl':f JJ and 34. leI V be Ihe Elldidean space R3 wilh Ih...
 5.3.34: III .11'1Lisl':f JJ and 34. leI V be Ihe Elldidean space R3 wilh Ih...
 5.3.35: Let A = [~ ~l Find a 2 x 2 matrix B of 0 such Ihnt A and B arc ort...
 5.3.36: Let V be the inner product space in Example 4. (a) [f p(t) = .Ji. f...
 5.3.37: Let C = [eii] be an II x II positive definite symmetric matrix and ...
 5.3.38: [f A and Bare n x n matrices. show that (A u. Bv) = (u. AT Bv) for ...
 5.3.39: [n the Euclidean space R" with the standard inner product. prove th...
 5.3.40: Consider Euclidean space Rl with the standard inner prodtlct and let
 5.3.41: Let V be an inner product space. Show that if v is orthogonal to WI...
 5.3.42: Suppose that (VI. V2 .... v,,) is an orthonormal set in R" with the...
 5.3.43: Suppose that (VI . V2 ..... v,,) is an orthogonal set In R" with th...
 5.3.44: [f A is nonsinguiar. prove that A T A is positive definite.
 5.3.45: If C is po~iti\'e definite. :lI1d x i= 0 is such that e x = kx 46....
 5.3.46: [f C is positive definite, show that its diagonal emries are positive.
 5.3.47: Let C be positive definite and r any scalar. Prove or disprove: rC ...
 5.3.48: If Band C are II x n positive definite mmrices. show that B + C is ...
 5.3.49: Let S be the set of /I x II positive definite matrices [s S a subsp...
 5.3.50: To compute the standard inner proouct of a pair of vectors u and V ...
 5.3.51: Exercise 41 in Section 5.1 can be generalized to R" . or even R" in...
 5.3.52: Exercise 41 in Section 5.1 can be generalized to R" . or even R" in...
 5.3.53: [f your software incorporates a computer algebm system that compute...
Solutions for Chapter 5.3: Inner Product Spaces
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780471669593
Solutions for Chapter 5.3: Inner Product Spaces
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. Chapter 5.3: Inner Product Spaces includes 53 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 53 problems in chapter 5.3: Inner Product Spaces have been answered, more than 9131 students have viewed full stepbystep solutions from this chapter. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780471669593.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

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

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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

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.

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).

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.

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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