 6.1: Let R4 have the Euclidean inner product. (a) Find a vector in R4 th...
 6.2: Prove: If u, v is the Euclidean inner product on Rn, and if A is an...
 6.3: LetM22 have the inner productU,V = tr(UTV ) = tr(V TU ) that was de...
 6.4: Let Ax = 0 be a system of m equations in n unknowns. Show that x = ...
 6.5: Use the CauchySchwarz inequality to show that if a1, a2,...,an are ...
 6.6: Show that if x and y are vectors in an inner product space and c is...
 6.7: Let R3 have the Euclidean inner product. Find two vectors of length...
 6.8: Find a weighted Euclidean inner product on Rn such that the vectors...
 6.9: Is there a weighted Euclidean inner product on R2 for which the vec...
 6.10: If u and v are vectors in an inner product space V, then u, v, and ...
 6.11: (a) As shown in Figure 3.2.6, the vectors (k, 0, 0), (0, k, 0), and...
 6.12: Let u and v be vectors in an inner product space. (a) Prove that u ...
 6.13: Let u be a vector in an inner product space V, and let {v1, v2,...,...
 6.14: Prove: If u, v1 and u, v2 are two inner products on a vector space ...
 6.15: Prove Theorem 6.2.5.
 6.16: Prove: If A has linearly independent column vectors, and if b is or...
 6.17: Is there any value of s for which x1 = 1 and x2 = 2 is the least sq...
 6.18: Show that if p and q are distinct positive integers, then the funct...
 6.19: Show that if p and q are positive integers, then the functions f(x)...
 6.20: Let W be the intersection of the planes x + y + z = 0 and x y + z =...
 6.21: Prove that if ad bc = 0, then the matrix A = a b c d has a unique Q...
Solutions for Chapter 6: Inner Product Spaces
Full solutions for Elementary Linear Algebra, Binder Ready Version: Applications Version  11th Edition
ISBN: 9781118474228
Solutions for Chapter 6: Inner Product Spaces
Get Full SolutionsSince 21 problems in chapter 6: Inner Product Spaces have been answered, more than 16908 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Linear Algebra, Binder Ready Version: Applications Version, edition: 11. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 6: Inner Product Spaces includes 21 full stepbystep solutions. Elementary Linear Algebra, Binder Ready Version: Applications Version was written by and is associated to the ISBN: 9781118474228.

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

Column space C (A) =
space of all combinations of the columns of A.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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

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.

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!).

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 .

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.

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

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

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

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.

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