 6.2.1: Let , , and have the Euclidean inner product. In each part, find th...
 6.2.2: Let have the inner product in Example 7 of Section 6.1 . Find the c...
 6.2.3: Let have the inner product in Example 6 of Section 6.1 . Find the c...
 6.2.4: In each part, determine whether the given vectors are orthogonal wi...
 6.2.5: Show that and are orthogonal with respect to the inner product in E...
 6.2.6: Let Which of the following matrices are orthogonal to A with respec...
 6.2.7: Do there exist scalars k and l such that the vectors , , and are mu...
 6.2.8: Let have the Euclidean inner product, and suppose that and . Find a...
 6.2.9: Let have the Euclidean inner product. For which values of k are u a...
 6.2.10: Let have the Euclidean inner product. Find two unit vectors that ar...
 6.2.11: In each part, verify that the CauchySchwarz inequality holds for th...
 6.2.12: In each part, verify that the CauchySchwarz inequality holds for th...
 6.2.13: Let have the Euclidean inner product, and let . Determine whether t...
 6.2.14: In Exercises 1415, assume that has the Euclidean inner product.Let ...
 6.2.15: In Exercises 1415, assume that has the Euclidean inner product.(a) ...
 6.2.16: Find a basis for the orthogonal complement of the subspace of spann...
 6.2.17: Let V be an inner product space. Show that if u and v are orthogona...
 6.2.18: Let V be an inner product space. Show that if w is orthogonal to bo...
 6.2.19: Let V be an inner product space. Show that if w is orthogonal to ea...
 6.2.20: Let be a basis for an inner product space V. Show that the zero vec...
 6.2.21: Let be a basis for a subspace W of V. Show that consists of all vec...
 6.2.22: Prove the following generalization of Theorem 6.2.3: If are pairwis...
 6.2.23: Prove: If u and v are matrices and A is an matrix, then
 6.2.24: Use the CauchySchwarz inequality to prove that for all real values ...
 6.2.25: Prove: If are positive real numbers, and if and are any two vectors...
 6.2.26: Show that equality holds in the CauchySchwarz inequality if and onl...
 6.2.27: Use vector methods to prove that a triangle that is inscribed in a ...
 6.2.28: As illustrated in the accompanying figure, the vectors and have nor...
 6.2.29: Calculus required Let and be continuous functions on . Prove: (a) (...
 6.2.30: Calculus required Let have the inner product and let . Show that if...
 6.2.31: (a) Let W be the line in an xycoordinate system in . Describe the ...
 6.2.32: Prove that Formula 4 holds for all nonzero vectors u and v in an in...
 6.2.a: In parts (a)(f) determine whether the statement is true or false, a...
 6.2.b: In parts (a)(f) determine whether the statement is true or false, a...
 6.2.c: In parts (a)(f) determine whether the statement is true or false, a...
 6.2.d: In parts (a)(f) determine whether the statement is true or false, a...
 6.2.e: In parts (a)(f) determine whether the statement is true or false, a...
 6.2.f: In parts (a)(f) determine whether the statement is true or false, a...
Solutions for Chapter 6.2: Inner Products
Full solutions for Elementary Linear Algebra: Applications Version  10th Edition
ISBN: 9780470432051
Solutions for Chapter 6.2: Inner Products
Get Full SolutionsSince 38 problems in chapter 6.2: Inner Products have been answered, more than 13825 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 6.2: Inner Products includes 38 full stepbystep solutions. Elementary Linear Algebra: Applications Version was written by and is associated to the ISBN: 9780470432051. This textbook survival guide was created for the textbook: Elementary Linear Algebra: Applications Version, edition: 10.

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

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

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

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

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

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.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = 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.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

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

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.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

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.