 4.4.1: 112 are about orthogonal vectors and orthogonal matrices.
 4.4.2: 112 are about orthogonal vectors and orthogonal matrices.
 4.4.3: 112 are about orthogonal vectors and orthogonal matrices.
 4.4.4: 112 are about orthogonal vectors and orthogonal matrices.
 4.4.5: 112 are about orthogonal vectors and orthogonal matrices.
 4.4.6: 112 are about orthogonal vectors and orthogonal matrices.
 4.4.7: 112 are about orthogonal vectors and orthogonal matrices.
 4.4.8: 112 are about orthogonal vectors and orthogonal matrices.
 4.4.9: 112 are about orthogonal vectors and orthogonal matrices.
 4.4.10: 112 are about orthogonal vectors and orthogonal matrices.
 4.4.11: 112 are about orthogonal vectors and orthogonal matrices.
 4.4.12: 112 are about orthogonal vectors and orthogonal matrices.
 4.4.13: 1325 are about the GramSchmidt process and A = QR.
 4.4.14: 1325 are about the GramSchmidt process and A = QR.
 4.4.15: 1325 are about the GramSchmidt process and A = QR.
 4.4.16: 1325 are about the GramSchmidt process and A = QR.
 4.4.17: 1325 are about the GramSchmidt process and A = QR.
 4.4.18: 1325 are about the GramSchmidt process and A = QR.
 4.4.19: 1325 are about the GramSchmidt process and A = QR.
 4.4.20: 1325 are about the GramSchmidt process and A = QR.
 4.4.21: 1325 are about the GramSchmidt process and A = QR.
 4.4.22: 1325 are about the GramSchmidt process and A = QR.
 4.4.23: 1325 are about the GramSchmidt process and A = QR.
 4.4.24: 1325 are about the GramSchmidt process and A = QR.
 4.4.25: 1325 are about the GramSchmidt process and A = QR.
 4.4.26: 2629 use the QR code in equations (1112). It executes GramSchmidt.
 4.4.27: 2629 use the QR code in equations (1112). It executes GramSchmidt.
 4.4.28: 2629 use the QR code in equations (1112). It executes GramSchmidt.
 4.4.29: 2629 use the QR code in equations (1112). It executes GramSchmidt.
 4.4.30: 3035 involve orthogonal matrices that are special.
 4.4.31: 3035 involve orthogonal matrices that are special.
 4.4.32: 3035 involve orthogonal matrices that are special.
 4.4.33: 3035 involve orthogonal matrices that are special.
 4.4.34: 3035 involve orthogonal matrices that are special.
 4.4.35: 3035 involve orthogonal matrices that are special.
 4.4.36: If A is m by n with rank n, qr(A) produces a square Q and zeros bel...
 4.4.37: We know that P = QQT is the projection onto the column space of Q(m...
Solutions for Chapter 4.4: Orthogonal Bases and GramSchmidt
Full solutions for Introduction to Linear Algebra  4th Edition
ISBN: 9780980232714
Solutions for Chapter 4.4: Orthogonal Bases and GramSchmidt
Get Full SolutionsThis textbook survival guide was created for the textbook: Introduction to Linear Algebra, edition: 4. Chapter 4.4: Orthogonal Bases and GramSchmidt includes 37 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Introduction to Linear Algebra was written by and is associated to the ISBN: 9780980232714. Since 37 problems in chapter 4.4: Orthogonal Bases and GramSchmidt have been answered, more than 28299 students have viewed full stepbystep solutions from this chapter.

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

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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

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.

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

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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

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 .

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

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).

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.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

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.

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

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