 9.3.1: For Exercises 14, find v given the coordinate vector vG withrespect...
 9.3.2: For Exercises 14, find v given the coordinate vector vG withrespect...
 9.3.3: For Exercises 14, find v given the coordinate vector vG withrespect...
 9.3.4: For Exercises 14, find v given the coordinate vector vG withrespect...
 9.3.5: For Exercises 512, find the coordinate vector of v with respect tot...
 9.3.6: For Exercises 512, find the coordinate vector of v with respect tot...
 9.3.7: For Exercises 512, find the coordinate vector of v with respect tot...
 9.3.8: For Exercises 512, find the coordinate vector of v with respect tot...
 9.3.9: For Exercises 512, find the coordinate vector of v with respect tot...
 9.3.10: For Exercises 512, find the coordinate vector of v with respect tot...
 9.3.11: For Exercises 512, find the coordinate vector of v with respect tot...
 9.3.12: For Exercises 512, find the coordinate vector of v with respect tot...
 9.3.13: For Exercises 1318, A is the matrix of linear transformationT : V W...
 9.3.14: For Exercises 1318, A is the matrix of linear transformationT : V W...
 9.3.15: For Exercises 1318, A is the matrix of linear transformationT : V W...
 9.3.16: For Exercises 1318, A is the matrix of linear transformationT : V W...
 9.3.17: For Exercises 1318, A is the matrix of linear transformationT : V W...
 9.3.18: For Exercises 1318, A is the matrix of linear transformationT : V W...
 9.3.19: For Exercises 1926, find the matrix Aof the linear transformationT ...
 9.3.20: For Exercises 1926, find the matrix Aof the linear transformationT ...
 9.3.21: For Exercises 1926, find the matrix Aof the linear transformationT ...
 9.3.22: For Exercises 1926, find the matrix Aof the linear transformationT ...
 9.3.23: For Exercises 1926, find the matrix Aof the linear transformationT ...
 9.3.24: For Exercises 1926, find the matrix Aof the linear transformationT ...
 9.3.25: For Exercises 1926, find the matrix Aof the linear transformationT ...
 9.3.26: For Exercises 1926, find the matrix Aof the linear transformationT ...
 9.3.27: For Exercises 2730, suppose thatA =abcde f is the matrix of T : V W...
 9.3.28: For Exercises 2730, suppose thatA =abcde f is the matrix of T : V W...
 9.3.29: For Exercises 2730, suppose thatA =abcde f is the matrix of T : V W...
 9.3.30: For Exercises 2730, suppose thatA =abcde f is the matrix of T : V W...
 9.3.31: Suppose that T : P1 P1 has matrix A =1 30 1withrespect to the basis...
 9.3.32: Suppose that T : P1 P1 has matrix A =2 71 3withrespect to the basis...
 9.3.33: Suppose that T : R2 P1 has matrix A =2 13 2withrespect to the basis...
 9.3.34: Suppose that T : R2 P1 has matrix A =1 31 4withrespect to the basis...
 9.3.35: FIND AN EXAMPLE For Exercises 3540, find an example thatmeets the g...
 9.3.36: FIND AN EXAMPLE For Exercises 3540, find an example thatmeets the g...
 9.3.37: FIND AN EXAMPLE For Exercises 3540, find an example thatmeets the g...
 9.3.38: FIND AN EXAMPLE For Exercises 3540, find an example thatmeets the g...
 9.3.39: FIND AN EXAMPLE For Exercises 3540, find an example thatmeets the g...
 9.3.40: FIND AN EXAMPLE For Exercises 3540, find an example thatmeets the g...
 9.3.41: TRUE OR FALSE For Exercises 4146, determine if the statementis true...
 9.3.42: TRUE OR FALSE For Exercises 4146, determine if the statementis true...
 9.3.43: TRUE OR FALSE For Exercises 4146, determine if the statementis true...
 9.3.44: TRUE OR FALSE For Exercises 4146, determine if the statementis true...
 9.3.45: TRUE OR FALSE For Exercises 4146, determine if the statementis true...
 9.3.46: TRUE OR FALSE For Exercises 4146, determine if the statementis true...
 9.3.47: Let G be a basis for a vector space V of dimension m. Showthat a se...
 9.3.48: LetG be a basis for a vector spaceV, and suppose that vG = wG.Prove...
 9.3.49: Let G be a basis for a vector space V of dimension m. Showthat a li...
 9.3.50: Suppose that Ais the matrix of linear transformation T : V W with r...
 9.3.51: Suppose that Ais the matrix of linear transformation T : V V with r...
Solutions for Chapter 9.3: The Matrix of a Linear Transformation
Full solutions for Linear Algebra with Applications  1st Edition
ISBN: 9780716786672
Solutions for Chapter 9.3: The Matrix of a Linear Transformation
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 1. Chapter 9.3: The Matrix of a Linear Transformation includes 51 full stepbystep solutions. Since 51 problems in chapter 9.3: The Matrix of a Linear Transformation have been answered, more than 4946 students have viewed full stepbystep solutions from this chapter. Linear Algebra with Applications was written by and is associated to the ISBN: 9780716786672.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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.

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

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].

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

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

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

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Iterative method.
A sequence of steps intended to approach the desired solution.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

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.

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

Solvable system Ax = b.
The right side b is in the column space of A.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
I don't want to reset my password
Need help? Contact support
Having trouble accessing your account? Let us help you, contact support at +1(510) 9441054 or support@studysoup.com
Forgot password? Reset it here