 4.1.1E: Let V be the first quadrant in the xyplane; that is, let a. If u a...
 4.1.2E: Let W be the union of the first and third quadrants in the xyplane...
 4.1.3E: Let H be the set of points inside and on the unit circle in the xy...
 4.1.4E: Construct a geometric figure that illustrates why a line in not thr...
 4.1.5E: In Exercises 5–8, determine if the given set is a subspace of for a...
 4.1.6E: In Exercises 5–8, determine if the given set is a subspace of for a...
 4.1.7E: In Exercises 5–8, determine if the given set is a subspace of for a...
 4.1.8E: In Exercises 5–8, determine if the given set is a subspace of Pn fo...
 4.1.9E: Let H be the set of all vectors of the form Find a vector v in such...
 4.1.10E: Let H be the set of all vectors of the form where t is any real num...
 4.1.11E: Let W be the set of all vectors of the form ,where b and c are arbi...
 4.1.12E: Let W be the set of all vectors of the form .Show that W is a subsp...
 4.1.13E: b. How many vectors are in Span ?c. Is w in the subspace spanned by...
 4.1.14E: Let be as in Exercise 13, and let .Is w in the subspace spanned by ...
 4.1.15E: In Exercises 15–18, let W be the set of all vectors of the form sho...
 4.1.16E: In Exercises 15–18, let W be the set of all vectors of the form sho...
 4.1.17E: In Exercises 15–18, let W be the set of all vectors of the form sho...
 4.1.18E: In Exercises 15–18, let W be the set of all vectors of the form sho...
 4.1.19E: If a mass m is placed at the end of a spring, and if the mass is pu...
 4.1.20E: The set of all continuous realvalued functions defined on a closed...
 4.1.21E: For fixed positive integers m and n, the set Mm × n of all m × n ma...
 4.1.22E: For fixed positive integers m and n, the set Mm × n of all m × n ma...
 4.1.23E: In Exercises 23 and 24, mark each statement True or False. Justify ...
 4.1.24E: In Exercises 23 and 24, mark each statement True or False. Justify ...
 4.1.25E: Exercises 25–29 show how the axioms for a vector space V can be use...
 4.1.26E: Exercises 25–29 show how the axioms for a vector space V can be use...
 4.1.27E: Exercises 25–29 show how the axioms for a vector space V can be use...
 4.1.28E: Exercises 25–29 show how the axioms for a vector space V can be use...
 4.1.29E: Exercises 25–29 show how the axioms for a vector space V can be use...
 4.1.30E: Suppose cu = 0 for some nonzero scalar c. Show that u = 0. Mention ...
 4.1.31E: Let u and v be vectors in a vector space V, and let H be any subspa...
 4.1.32E: Let H and K be subspaces of a vector space V. The intersection of H...
 4.1.33E: Given subspaces H and K of a vector space V , the sum of H and K, w...
 4.1.34E: Suppose are vectors in a vector space V , and letShow that
 4.1.35E: [M] Show that w is in the subspace of spanned by
 4.1.36E: [M] Determine if y is in the subspace of spanned by the columns of ...
 4.1.37E: [M] The vector spacecontains at least two interesting functions tha...
 4.1.38E: [M] Repeat Exercise 37 for the functionsfghIn the vector space Span...
Solutions for Chapter 4.1: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 4.1
Get Full SolutionsChapter 4.1 includes 38 full stepbystep solutions. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4. Since 38 problems in chapter 4.1 have been answered, more than 35391 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

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.

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

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

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

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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.

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

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.