 6.1.1: Label the following statements as true or false. (a) An inner produ...
 6.1.2: Let. x = (2.1 f i,i) and y  (2 v',2. 1 42/) be vectors in C3 . Co...
 6.1.3: In C([0, 1]), let f(t)  I and g(t) = e'. Compute (f\g) (as defined...
 6.1.4: (a) Complete the proof in Example 5 that (, ) is an inner product (...
 6.1.5: In C2 . show that (x.y) xAy* is an inner product, where .4  1 i i...
 6.1.6: Complete the proof of Theorem 6.1.
 6.1.7: Complete the proof of Theorem Ci.2
 6.1.8: Provide reasons why each of the following is not an inner product o...
 6.1.9: Let 0 be a basis for a finitedimensional inner product space. (a) ...
 6.1.10: Let V be an inner product space, and suppose that X and y are ortho...
 6.1.11: Prove the parallelogram law on an inner product space V: that is, s...
 6.1.12: Let {v\. v2,...,
 6.1.13: Suppose that (. )x and (, )., are two inner products on a vector sp...
 6.1.14: Let A and B be n x n matrices, and let c be a scalar. Prove that (A...
 6.1.15: (a) Prove that if V is an inner product space, then  (x.y)  = r...
 6.1.16: (a) Show that the vector space H with (, ) defined on page 332 is a...
 6.1.17: Let T be a linear operator on an inner product space V. and suppose...
 6.1.18: Let V be a vector space over /'. where F = R or F  ('. and let W b...
 6.1.19: Let V be an inner product space. Prove that (a) \\x y\\2 = \\x\\2 2...
 6.1.20: Let V be an inner product space over F. Prove the polar identities:...
 6.1.21: Let A be an n x n matrix. Define Ai = (A + A*) and A2 = (AA*). 2...
 6.1.22: Let V be a real or complex vector space (possibly infinitedimensio...
 6.1.23: Let V = Fn , and let A E M.X/I(/"). (a) Prove that (x. Ay) = (A*x,y...
 6.1.24: The following definition is used in Exercises 24 27. Definition. Le...
 6.1.25: The following definition is used in Exercises 24 27. Definition. Le...
 6.1.26: The following definition is used in Exercises 24 27. Definition. Le...
 6.1.27: The following definition is used in Exercises 24 27. Definition. Le...
 6.1.28: Let V be a complex inner product space with an inner product (, ). ...
 6.1.29: Let V be a complex inner product space with an inner product (, ). ...
 6.1.30: Let   be a norm (as defined in Exercise 24) on a complex vector...
Solutions for Chapter 6.1: Inner Products and Norms
Full solutions for Linear Algebra  4th Edition
ISBN: 9780130084514
Solutions for Chapter 6.1: Inner Products and Norms
Get Full SolutionsSince 30 problems in chapter 6.1: Inner Products and Norms have been answered, more than 10959 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra was written by and is associated to the ISBN: 9780130084514. Chapter 6.1: Inner Products and Norms includes 30 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra , edition: 4.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

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

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

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).

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].

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

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

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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.

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.

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

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).

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.