 10.1.1: For Exercises 18, compute the indicated inner product.u, v for u =1...
 10.1.2: For Exercises 18, compute the indicated inner product.u, v for u =2...
 10.1.3: For Exercises 18, compute the indicated inner product.p, qfor p(x) ...
 10.1.4: For Exercises 18, compute the indicated inner product.. p, qfor p(x...
 10.1.5: For Exercises 18, compute the indicated inner product. f, g for f (...
 10.1.6: For Exercises 18, compute the indicated inner product. f, g for f (...
 10.1.7: For Exercises 18, compute the indicated inner product.A, B = tr(AT ...
 10.1.8: For Exercises 18, compute the indicated inner product.A, B = tr(AT ...
 10.1.9: Suppose that u = 101and v =212are orthogonal withrespect to the inn...
 10.1.10: Suppose that u =322and v = 512are orthogonal withrespect to the inn...
 10.1.11: Suppose that p(x) = x + 2, q(x) = 3x + 1 are orthogonalwith respect...
 10.1.12: Suppose that p(x) = x23x1, q(x) = x+2 are orthogonalwith respect to...
 10.1.13: Suppose that f (x) = 2x and g (x) = x +b. For what value(s)of b are...
 10.1.14: Suppose that f (x) = x2 and g (x) = x + b. For what value(s)of b ar...
 10.1.15: For Exercises 1522, compute the norm with respect to the indicatedi...
 10.1.16: For Exercises 1522, compute the norm with respect to the indicatedi...
 10.1.17: For Exercises 1522, compute the norm with respect to the indicatedi...
 10.1.18: For Exercises 1522, compute the norm with respect to the indicatedi...
 10.1.19: For Exercises 1522, compute the norm with respect to the indicatedi...
 10.1.20: For Exercises 1522, compute the norm with respect to the indicatedi...
 10.1.21: For Exercises 1522, compute the norm with respect to the indicatedi...
 10.1.22: For Exercises 1522, compute the norm with respect to the indicatedi...
 10.1.23: For Exercises 2330, compute the indicated projection with respectto...
 10.1.24: For Exercises 2330, compute the indicated projection with respectto...
 10.1.25: For Exercises 2330, compute the indicated projection with respectto...
 10.1.26: For Exercises 2330, compute the indicated projection with respectto...
 10.1.27: For Exercises 2330, compute the indicated projection with respectto...
 10.1.28: For Exercises 2330, compute the indicated projection with respectto...
 10.1.29: For Exercises 2330, compute the indicated projection with respectto...
 10.1.30: For Exercises 2330, compute the indicated projection with respectto...
 10.1.31: FIND AN EXAMPLE For Exercises 3140, find an example thatmeets the g...
 10.1.32: FIND AN EXAMPLE For Exercises 3140, find an example thatmeets the g...
 10.1.33: FIND AN EXAMPLE For Exercises 3140, find an example thatmeets the g...
 10.1.34: FIND AN EXAMPLE For Exercises 3140, find an example thatmeets the g...
 10.1.35: FIND AN EXAMPLE For Exercises 3140, find an example thatmeets the g...
 10.1.36: FIND AN EXAMPLE For Exercises 3140, find an example thatmeets the g...
 10.1.37: FIND AN EXAMPLE For Exercises 3140, find an example thatmeets the g...
 10.1.38: FIND AN EXAMPLE For Exercises 3140, find an example thatmeets the g...
 10.1.39: FIND AN EXAMPLE For Exercises 3140, find an example thatmeets the g...
 10.1.40: FIND AN EXAMPLE For Exercises 3140, find an example thatmeets the g...
 10.1.41: TRUE OR FALSE For Exercises 4150, determine if the statementis true...
 10.1.42: TRUE OR FALSE For Exercises 4150, determine if the statementis true...
 10.1.43: TRUE OR FALSE For Exercises 4150, determine if the statementis true...
 10.1.44: TRUE OR FALSE For Exercises 4150, determine if the statementis true...
 10.1.45: TRUE OR FALSE For Exercises 4150, determine if the statementis true...
 10.1.46: TRUE OR FALSE For Exercises 4150, determine if the statementis true...
 10.1.47: TRUE OR FALSE For Exercises 4150, determine if the statementis true...
 10.1.48: TRUE OR FALSE For Exercises 4150, determine if the statementis true...
 10.1.49: TRUE OR FALSE For Exercises 4150, determine if the statementis true...
 10.1.50: TRUE OR FALSE For Exercises 4150, determine if the statementis true...
 10.1.51: Complete Example 1. Prove that the weighted dot product onRn given ...
 10.1.52: Complete Example 3. Show that properties (a)(c) of Definition10.1 a...
 10.1.53: A weighted version of the inner product given in Example 3is define...
 10.1.54: Let f and g be continuous functions in C[1, 1]. Show thatthe weight...
 10.1.55: Let f and g be continuous functions in C[, ]. Show that f, g =1 f (...
 10.1.56: Complete the proof of Theorem 10.6, by showing that parts(c) and (d...
 10.1.57: Prove that A, B = tr(AT B) is an inner product on R33.
 10.1.58: For nonzero vectors u and v, show that there is equality in theCauc...
 10.1.59: For Exercises 5968, u, v, and w (and their subscripted associates)a...
 10.1.60: For Exercises 5968, u, v, and w (and their subscripted associates)a...
 10.1.61: For Exercises 5968, u, v, and w (and their subscripted associates)a...
 10.1.62: For Exercises 5968, u, v, and w (and their subscripted associates)a...
 10.1.63: For Exercises 5968, u, v, and w (and their subscripted associates)a...
 10.1.64: For Exercises 5968, u, v, and w (and their subscripted associates)a...
 10.1.65: For Exercises 5968, u, v, and w (and their subscripted associates)a...
 10.1.66: For Exercises 5968, u, v, and w (and their subscripted associates)a...
 10.1.67: For Exercises 5968, u, v, and w (and their subscripted associates)a...
 10.1.68: For Exercises 5968, u, v, and w (and their subscripted associates)a...
 10.1.69: If S is a subspace of a finitedimensional inner product space V, a...
 10.1.70: If S is a subspace of a finitedimensional inner product space V, a...
 10.1.71: If S is a subspace of a finitedimensional inner product space V, a...
 10.1.72: If S is a subspace of a finitedimensional inner product space V, a...
Solutions for Chapter 10.1: Inner Products
Full solutions for Linear Algebra with Applications  1st Edition
ISBN: 9780716786672
Solutions for Chapter 10.1: Inner Products
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780716786672. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 1. Chapter 10.1: Inner Products includes 72 full stepbystep solutions. Since 72 problems in chapter 10.1: Inner Products have been answered, more than 14619 students have viewed full stepbystep solutions from this chapter.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

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).

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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

Solvable system Ax = b.
The right side b is in the column space of A.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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