 2.3.1: For each ofrhe mmrices A and Bin Exercises 1 8, derennine whether ...
 2.3.2: For each ofrhe mmrices A and Bin Exercises 1 8, derennine whether ...
 2.3.3: For each ofrhe mmrices A and Bin Exercises 1 8, derennine whether ...
 2.3.4: For each ofrhe mmrices A and Bin Exercises 1 8, derennine whether ...
 2.3.5: For each ofrhe mmrices A and Bin Exercises 1 8, derennine whether ...
 2.3.6: For each ofrhe mmrices A and Bin Exercises 1 8, derennine whether ...
 2.3.7: For each ofrhe mmrices A and Bin Exercises 1 8, derennine whether ...
 2.3.8: For each ofrhe mmrices A and Bin Exercises 1 8, derennine whether ...
 2.3.9: For Exercises 9 14, find file value of each malrix expression. whe...
 2.3.10: For Exercises 9 14, find file value of each malrix expression. whe...
 2.3.11: For Exercises 9 14, find file value of each malrix expression. whe...
 2.3.12: For Exercises 9 14, find file value of each malrix expression. whe...
 2.3.13: For Exercises 9 14, find file value of each malrix expression. whe...
 2.3.14: For Exercises 9 14, find file value of each malrix expression. whe...
 2.3.15: In Exercises 15 22, find file inverse of each elemelllaty malrix. :
 2.3.16: In Exercises 15 22, find file inverse of each elemelllaty malrix. ...
 2.3.17: In Exercises 15 22, find file inverse of each elemelllaty malrix. ...
 2.3.18: In Exercises 15 22, find file inverse of each elemelllaty malrix. ...
 2.3.19: In Exercises 15 22, find file inverse of each elemelllaty malrix. ...
 2.3.20: In Exercises 15 22, find file inverse of each elemelllaty malrix. ...
 2.3.21: In Exercises 15 22, find file inverse of each elemelllaty malrix.1...
 2.3.22: In Exercises 15 22, find file inverse of each elemelllaty malrix.1...
 2.3.23: In Exercises 23 32, find tm elementary nwtri.x E such film EA = B ...
 2.3.24: In Exercises 23 32, find tm elementary nwtri.x E such film EA = B ...
 2.3.25: In Exercises 23 32, find tm elementary nwtri.x E such film EA = B ...
 2.3.26: In Exercises 23 32, find tm elementary nwtri.x E such film EA = B ...
 2.3.27: In Exercises 23 32, find tm elementary nwtri.x E such film EA = B ...
 2.3.28: In Exercises 23 32, find tm elementary nwtri.x E such film EA = B ...
 2.3.29: In Exercises 23 32, find tm elementary nwtri.x E such film EA = B i
 2.3.30: In Exercises 23 32, find tm elementary nwtri.x E such film EA = B ...
 2.3.31: In Exercises 23 32, find tm elementary nwtri.x E such film EA = B ...
 2.3.32: In Exercises 23 32, find tm elementary nwtri.x E such film EA = B ...
 2.3.33: Every square matrix is invenible.
 2.3.34: Invertible matrices arc square.
 2.3.35: Elementary matrices are invertible.
 2.3.36: If A and 8 are matrices such that AB = !,. for some 11, then both A...
 2.3.37: If 8 and C are inverses of a matrix A, then 8 =C.
 2.3.38: f A and 8 are invenible n x 11 matrices, then AB T is invertible.
 2.3.39: An invertible matrix may have more than one inverse.
 2.3.40: For any matrices A and 8, if A is the inverse of 8 , then 8 is the ...
 2.3.41: For any matrice.~ A and 8 , if A is the inverse of Br. then A is th...
 2.3.42: If A and 8 are invertible 11 x 11 matrices, then AB is also inve1ti...
 2.3.43: If A and 8 are invertible 11 x n matrices, then (AB) 1 = A 18  1
 2.3.44: An e lementary matrix is a matrix that can be obtai ned by a sequen...
 2.3.45: An elementary 11 x 11 matrix has at most 11 + I nonzero entries
 2.3.46: An elementary 11 x 11 matrix has at most 11 + I nonzero entries
 2.3.47: Every elementary matrix is invertible.
 2.3.48: If A and 8 are m x 11 matrices and 8 can be obtained from A by an e...
 2.3.49: If R is the reduced row echelon form of a matrix A, then there exis...
 2.3.50: Let R be the reduced row echelon form of a matrix A. If colunm j of...
 2.3.51: The pivot columns of a matrix are linearly dependent.
 2.3.52: Every column of a matrL' is a linear combination of its pivot columns
 2.3.53: 3 Let A., be the arotation matrix, Prove that (A,)T = (A,)1
 2.3.54: Let A = [: ~ l (a) Suppose ad  be # 0. and 8 I [ d  ad be c ...
 2.3.55: Prove that the product of e lementary matrices is invertible
 2.3.56: (a) Let A be an invertible 11 x 11 matrix, and let u and v be vecto...
 2.3.57: Let Q be an invertible 11. x 11 matrix, Prove that the subset {u1, ...
 2.3.58: Prove Theorem 2.2(a).
 2.3.59: Prove that if A, 8, and C are invertible 11 x 11 matrices, then ABC...
 2.3.60: Let A and 8 be n x 11 matrices such that both A and AB are invertib...
 2.3.61: Let A and 8 be 11 x 11 matrices such that AB = 1,.. Prove that the ...
 2.3.62: Prove that if A is an 111 x 11 matrix and 8 is an 11 x p matrix, th...
 2.3.63: Prove that if 8 is an 11 x 11 matrix with rank 11 , then there exis...
 2.3.64: Prove that if A and 8 arc 11 x 11 matrices such that A8 = 1,., then...
 2.3.65: Prove that if an 11 x 11 matrix has rank 11, then it is invertible....
 2.3.66: Let M = 02 8 , where A and 8 are square and Ot ;md 0 2 arc zero mat...
 2.3.67: /11 Exercises 67 74, find the matrix A, given the retluced row ech...
 2.3.68: /11 Exercises 67 74, find the matrix A, given the retluced row ech...
 2.3.69: /11 Exercises 67 74, find the matrix A, given the retluced row ech...
 2.3.70: /11 Exercises 67 74, find the matrix A, given the retluced row ech...
 2.3.71: /11 Exercises 67 74, find the matrix A, given the retluced row ech...
 2.3.72: /11 Exercises 67 74, find the matrix A, given the retluced row ech...
 2.3.73: /11 Exercises 67 74, find the matrix A, given the retluced row ech...
 2.3.74: /11 Exercises 67 74, find the matrix A, given the retluced row ech...
 2.3.75: In Exercises 7578. write the indicated colwnn of A=[~ 2 1 4 I I...
 2.3.76: In Exercises 7578. write the indicated colwnn of A=[~ 2 1 4 I I...
 2.3.77: In Exercises 7578. write the indicated colwnn of A=[~ 2 1 4 I I...
 2.3.78: In Exercises 7578. write the indicated colwnn of A=[~ 2 1 4 I I...
 2.3.79: /11 E.1erciw,, 7982, urite the indicmed wi1111111 of 8 ~ [ ; 0 I ...
 2.3.80: /11 E.1erciw,, 7982, urite the indicmed wi1111111 of 8 ~ [ ; 0 I ...
 2.3.81: /11 E.1erciw,, 7982, urite the indicmed wi1111111 of 8 ~ [ ; 0 I ...
 2.3.82: /11 E.1erciw,, 7982, urite the indicmed wi1111111 of 8 ~ [ ; 0 I ...
 2.3.83: Suppose that u and v are linearly independent vectors in 'R3. Find ...
 2.3.84: Let A be an " x n invenible matrix, and let CJ be the jth standard ...
 2.3.85: Let A be a matrix with reduced row echelon form R. Usc the column c...
 2.3.86: Let R be an 111 x 11 matrix in reduced row echelon form. Find a rel...
 2.3.87: Let R be an 111 x 11 matrix in reduced row echelon form with rank R...
 2.3.88: Let A be an m x n matrix with reduced row echelon form R. Then ther...
 2.3.89: Let A and 8 be 111 x 11 matrices. Prove that the following ccmditio...
 2.3.90: Let A be an 11 x 11 matrix. Find a property of A that i' equivalent...
 2.3.91: Let A be a 2 x 3 matrix, and let E be an elementary matrix obtained...
 2.3.92: Let A and 8 be m x 11 matrices. Prove that the following conditions...
 2.3.93: Let f:.' be an 11 x 11 matrix. Prove that E is an elementary matrix...
 2.3.94: Prove that if tt matrix E i> oblinglc e lementary column operation,...
 2.3.95: In E.urift'f 95 99. liSt' eithu a calculator with lllfltrit rapa bi...
 2.3.96: In E.urift'f 95 99. liSt' eithu a calculator with lllfltrit rapa bi...
 2.3.97: In E.urift'f 95 99. liSt' eithu a calculator with lllfltrit rapa bi...
 2.3.98: In E.urift'f 95 99. liSt' eithu a calculator with lllfltrit rapa bi...
 2.3.99: In E.urift'f 95 99. liSt' eithu a calculator with lllfltrit rapa bi...
Solutions for Chapter 2.3: MATRICES AND LINEAR TRANSFORMATIONS
Full solutions for Elementary Linear Algebra: A Matrix Approach  2nd Edition
ISBN: 9780131871410
Solutions for Chapter 2.3: MATRICES AND LINEAR TRANSFORMATIONS
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Linear Algebra: A Matrix Approach, edition: 2. Chapter 2.3: MATRICES AND LINEAR TRANSFORMATIONS includes 99 full stepbystep solutions. Since 99 problems in chapter 2.3: MATRICES AND LINEAR TRANSFORMATIONS have been answered, more than 23081 students have viewed full stepbystep solutions from this chapter. Elementary Linear Algebra: A Matrix Approach was written by and is associated to the ISBN: 9780131871410. This expansive textbook survival guide covers the following chapters and their solutions.

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.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

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

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

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.

Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.

Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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

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.