 1.10.1E: The container of a breakfast cereal usually lists the number of cal...
 1.10.2E: One serving of Shredded Wheat supplies 160 calories, 5 g of protein...
 1.10.3E: After taking a nutrition class, a big Annie’s® Mac and Cheese fan d...
 1.10.4E: The Cambridge Diet supplies .8 g of calcium per day, in addition to...
 1.10.5E: In Exercises 5–8, write a matrix equation that determines the loop ...
 1.10.6E: In Exercises 5–8, write a matrix equation that determines the loop ...
 1.10.7E: In Exercises 5–8, write a matrix equation that determines the loop ...
 1.10.8E: In Exercises 5–8, write a matrix equation that determines the loop ...
 1.10.9E: In a certain region, about 7% of a city’s population moves to the s...
 1.10.10E: In a certain region, about 6% of a city’s population moves to the s...
 1.10.11E: In 1994, the population of California was 31,524,000, and the popul...
 1.10.12E: [M] Budget® Rent A Car in Wichita, Kansas has a fleet of about 500 ...
 1.10.13E: [M] Let M and x0 be as in Example 3.a. Compute the population vecto...
 1.10.14E: [M] Study how changes in boundary temperatures on a steel plate aff...
Solutions for Chapter 1.10: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 1.10
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.10 includes 14 full stepbystep solutions. Since 14 problems in chapter 1.10 have been answered, more than 30260 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.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Column space C (A) =
space of all combinations of the columns of A.

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.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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

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

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

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

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

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

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.