 4.10.1: Consider the following systems of linear equations. For convenience...
 4.10.2: Consider the following systems of linear equations. For convenience...
 4.10.3: Consider the following systems of linear equations. For convenience...
 4.10.4: Consider the following systems of linear equations. For convenience...
 4.10.5: Consider the following systems of linear equations. For convenience...
 4.10.6: Consider the following systems of linear equations. For convenience...
 4.10.7: Considerthe transformationT: R 3 R 3 defined bythe following matrix...
 4.10.8: In this section we solved nonhomogeneous systems of equations, arri...
 4.10.9: Determine a vector other than (S, 2, 0, 0) that can be used as a pa...
 4.10.10: Construct a nonhomogeneous system of linear equations Ax = y that h...
 4.10.11: Construct a nonhomogeneous system of linear equations Ax = y that h...
 4.10.12: Prove that the solution to a nonhomogeneous system of linear equati...
 4.10.13: Exercises 13 and 14 are intended for students who have a knowledge ...
 4.10.14: Exercises 13 and 14 are intended for students who have a knowledge ...
Solutions for Chapter 4.10: Transformations and Systems of Linear Equations
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9781449679545
Solutions for Chapter 4.10: Transformations and Systems of Linear Equations
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 14 problems in chapter 4.10: Transformations and Systems of Linear Equations have been answered, more than 8491 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Linear Algebra with Applications was written by and is associated to the ISBN: 9781449679545. Chapter 4.10: Transformations and Systems of Linear Equations includes 14 full stepbystep solutions.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

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

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).

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.

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

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

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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

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 .

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

Outer product uv T
= column times row = rank one matrix.

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

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.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

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

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