 4.1.1: Show that each of the following are linear operators on R2. Describ...
 4.1.2: Let L be the linear operator on R2 defined by L(x) = (x1 cos x2 sin...
 4.1.3: Let a be a fixed nonzero vector in R2. A mapping of the form L(x) =...
 4.1.4: Let L : R2 R2 be a linear operator. If L((1, 2)T ) = (2, 3)T and L(...
 4.1.5: Determine whether the following are linear transformations from R3 ...
 4.1.6: Determine whether the following are linear transformations from R2 ...
 4.1.7: Determine whether the following are linear operators on Rnn: (a) L(...
 4.1.8: Let C be a fixed n n matrix. Determine whether the following are li...
 4.1.9: Determine whether the following are linear transformations from P2 ...
 4.1.10: For each f C[0, 1], define L( f ) = F, where F(x) = _ x 0 f (t) dt ...
 4.1.11: Determine whether the following are linear transformations from C[0...
 4.1.12: Use mathematical induction to prove that if L is a linear transform...
 4.1.13: Let {v1, . . . , vn} be a basis for a vector space V, and let L1 an...
 4.1.14: Let L be a linear operator on R1 and let a = L(1). Show that L(x) =...
 4.1.15: Let L be a linear operator on a vector space V. Define Ln, n 1, rec...
 4.1.16: Let L1 : U V and L2 : V W be linear transformations, and let L = L2...
 4.1.17: Determine the kernel and range of each of the following linear oper...
 4.1.18: Let S be the subspace of R3 spanned by e1 and e2. For each linear o...
 4.1.19: Find the kernel and range of each of the following linear operators...
 4.1.20: Let L : V W be a linear transformation, and let T be a subspace of ...
 4.1.21: A linear transformation L : V W is said to be onetoone if L(v1) =...
 4.1.22: A linear transformation L : V W is said to map V onto W if L(V) = W...
 4.1.23: Which of the operators defined in Exercise 17 are onetoone? Which...
 4.1.24: Let A be a 2 2 matrix, and let L A be the linear operator defined b...
 4.1.25: Let D be the differentiation operator on P3, and let S = {p P3  p(...
Solutions for Chapter 4.1: Definition and Examples
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9780136009290
Solutions for Chapter 4.1: Definition and Examples
Get Full SolutionsChapter 4.1: Definition and Examples includes 25 full stepbystep solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009290. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Since 25 problems in chapter 4.1: Definition and Examples have been answered, more than 5060 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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.

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.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

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

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.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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.

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!

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.

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

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

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.