 5.2.1: Using paper and pencil, perform the GramSchmidt process on the seq...
 5.2.2: Using paper and pencil, perform the GramSchmidt process on the seq...
 5.2.3: Using paper and pencil, perform the GramSchmidt process on the seq...
 5.2.4: Using paper and pencil, perform the GramSchmidt process on the seq...
 5.2.5: Using paper and pencil, perform the GramSchmidt process on the seq...
 5.2.6: Using paper and pencil, perform the GramSchmidt process on the seq...
 5.2.7: Using paper and pencil, perform the GramSchmidt process on the seq...
 5.2.8: Using paper and pencil, perform the GramSchmidt process on the seq...
 5.2.9: Using paper and pencil, perform the GramSchmidt process on the seq...
 5.2.10: Using paper and pencil, perform the GramSchmidt process on the seq...
 5.2.11: Using paper and pencil, perform the GramSchmidt process on the seq...
 5.2.12: Using paper and pencil, perform the GramSchmidt process on the seq...
 5.2.13: Using paper and pencil, perform the GramSchmidt process on the seq...
 5.2.14: Using paper and pencil, perform the GramSchmidt process on the seq...
 5.2.15: Using paper and pencil, find the QR factorizations of the matrices ...
 5.2.16: Using paper and pencil, find the QR factorizations of the matrices ...
 5.2.17: Using paper and pencil, find the QR factorizations of the matrices ...
 5.2.18: Using paper and pencil, find the QR factorizations of the matrices ...
 5.2.19: Using paper and pencil, find the QR factorizations of the matrices ...
 5.2.20: Using paper and pencil, find the QR factorizations of the matrices ...
 5.2.21: Using paper and pencil, find the QR factorizations of the matrices ...
 5.2.22: Using paper and pencil, find the QR factorizations of the matrices ...
 5.2.23: Using paper and pencil, find the QR factorizations of the matrices ...
 5.2.24: Using paper and pencil, find the QR factorizations of the matrices ...
 5.2.25: Using paper and pencil, find the QR factorizations of the matrices ...
 5.2.26: Using paper and pencil, find the QR factorizations of the matrices ...
 5.2.27: Using paper and pencil, find the QR factorizations of the matrices ...
 5.2.28: Using paper and pencil, find the QR factorizations of the matrices ...
 5.2.29: Perform the GramSchmidt process on the following basis of M2:'3" ...
 5.2.30: Consider two linearly independent vectors v\ =and vi = in IR2. Draw...
 5.2.31: Perform the GramSchmidt process on the following basis ofR3:v\ =He...
 5.2.32: ind an orthonormal basis of the plane*1 + *2 + *3 = 0
 5.2.33: Find an orthonormal basis of the kernel of the matrix. fl 1 1 1 b
 5.2.34: Find an orthonormal basis of the kernel of the matrix 1 1 1 1 ' 1 2...
 5.2.35: Find an orthonormal basis of the image of the matrix 'l 2 f
 5.2.36: Consider the matrix _1 1 1  1 1 "I 1 11 21  1 1  13 4 0Find th...
 5.2.37: Consider the matrixl l l l " "3 4' 1 1 11 1 0 5 2 11 1  1 0 0 1...
 5.2.38: Find the QR factorization of "0 3A =0 0 0
 5.2.39: Find an orthonormal basis u \, u2, m3 of R3 such that span(Mi) = sp...
 5.2.40: Consider an invertible n x n matrix A whose columns are orthogonal,...
 5.2.41: Consider an invertible upper triangular n x n matrix A. What does t...
 5.2.42: The two column vectors v\ and v2 of a 2 x 2 matrix 4 are shown in t...
 5.2.43: Consider a block matrixA = [ A i A2]with linearly independent colum...
 5.2.44: Consider an n x m matrix A with rank(y4) < m. Is it always possible...
 5.2.45: Consider an n x m matrix A with rank (A) = m. Is it always possible...
Solutions for Chapter 5.2: GramSchmidt Process and QR Factorization
Full solutions for Linear Algebra with Applications  4th Edition
ISBN: 9780136009269
Solutions for Chapter 5.2: GramSchmidt Process and QR Factorization
Get Full SolutionsLinear Algebra with Applications was written by and is associated to the ISBN: 9780136009269. Chapter 5.2: GramSchmidt Process and QR Factorization includes 45 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 45 problems in chapter 5.2: GramSchmidt Process and QR Factorization have been answered, more than 15362 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 4.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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].

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = 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 .

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

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

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