 4.4.1: In Exercises 16, the 8coordinate vector of\' is gien. Find v if 1...
 4.4.2: In Exercises 16, the 8coordinate vector of\' is gien. Find v if 2...
 4.4.3: In Exercises 16, the 8coordinate vector of\' is gien. Find v if 3...
 4.4.4: In Exercises 16, the 8coordinate vector of\' is gien. Find v if 4...
 4.4.5: In Exercises 16, the 8coordinate vector of\' is gien. Find v if 5...
 4.4.6: In Exercises 16, the 8coordinate vector of\' is gien. Find v if 6...
 4.4.7: In Exercises 7 10, the 8coordinate vector of v is given. Find v i...
 4.4.8: In Exercises 7 10, the 8coordinate vector of v is given. Find v i...
 4.4.9: In Exercises 7 10, the 8coordinate vector of v is given. Find v i...
 4.4.10: In Exercises 7 10, the 8coordinate vector of v is given. Find v i...
 4.4.11: (a) Prove that 8 = {[~]. [ ~] } is a basis for R.2. (b) Find [vJ 6...
 4.4.12: (a) Prove that 8 = {[ _;J, ~] is a basis for R.2 . (b) Fmd . [v] ....
 4.4.13: (a) Prove that 8= {[ ~l [ _:]. [ l]} is a basis for nJ (b) Find (...
 4.4.14: . (a) Provethat8= { [ :][ :]. [:] } isabasisfor nJ. (b) Find [v[G...
 4.4.15: In Exercises 15 18, a vector is gien. Find its 8coordinate vector...
 4.4.16: In Exercises 15 18, a vector is gien. Find its 8coordinate vector...
 4.4.17: In Exercises 15 18, a vector is gien. Find its 8coordinate vector...
 4.4.18: In Exercises 15 18, a vector is gien. Find its 8coordinate vector...
 4.4.19: !11 Exercises 19 22. aecTor is given Find iTs 6coottlinme vector ...
 4.4.20: !11 Exercises 19 22. aecTor is given Find iTs 6coottlinme vector ...
 4.4.21: !11 Exercises 19 22. aecTor is given Find iTs 6coottlinme vector ...
 4.4.22: !11 Exercises 19 22. aecTor is given Find iTs 6coottlinme vector ...
 4.4.23: Find the unique representation of u = ~ J as a linear combination of
 4.4.24: Find the unique represcnt<tion of u = ~ J as a linear combination of
 4.4.25: Find the unique representation of u = [ ~ J as a linear combination of
 4.4.26: Find the unique representation of u = [ ~] as a linear combination of
 4.4.27: Find the unique representation of 11 = [ ~] as a linear combination of
 4.4.28: Find the unique representution of 11 = [ ~] as a linear combination of
 4.4.29: Find the unique representation of 11 = [ ~] as a linear combination of
 4.4.30: Find the unique representation of u = [ ~] as a linear combination of
 4.4.31: If S is a generating set for a subspace V of n". then every vector ...
 4.4.32: The components of the 8coordinate vector of u are the coefficients...
 4.4.33: If[; is the standard basis for n. then [v)c = v for all v in 'R.".
 4.4.34: The 8coordinate vector of a vector in B is a standard vector.
 4.4.35: If B is a basis for n" and B is the matrix whose columns are the ve...
 4.4.36: If B = lb 1, bz, ... , b,l is any basis for n", then, for every vec...
 4.4.37: If the columns of an 11 X II matrix form a basis fOI' n". then the ...
 4.4.38: If B is a matrix whose columns are the vectors in a basis for n", t...
 4.4.39: If 8 is a matrix whose columns arc the vectors in a basis 8 for n. ...
 4.4.40: If 8 is a matrix whose columns are the vectors in a basis B for n",...
 4.4.41: If 8 is any basis for n", then 1016 = 0
 4.4.42: If u and v are any vectors in n and B is any basis for n". then (u ...
 4.4.43: If v is ;my vector in n. B is any basis for n". and c is a scalar, ...
 4.4.44: Suppose th
 4.4.45: Suppose that the x' , y' axes are obtained by rotating the usual ...
 4.4.46: lf Ao is a rotation matrix, then A'/; = A0 1. (x')
 4.4.47: The graph of an equation of the form 7 + ;r = I is an ellipse.
 4.4.48: The graph of an equation of the form .:....,   , = I a b is a p...
 4.4.49: The graph of an ellipse with center at the origin can be (x')2 (y')...
 4.4.50: The graph of a hyperbola with center at the origi n can (x')2 (y')2...
 4.4.51: Let B = lb , b2J. where bt = ~] and bz = [il (a) Show that B is a b...
 4.4.52: Let B = (b 1 b2l where bt = ~] and b2 = [ ~l (a) Show that l3 is a...
 4.4.53: Let B=(b,, b2, bJ}. where bt = [ _:J b2= [=:J and bJ = [ =n (a) Sho...
 4.4.54: Let B= (b1 . b2. bJ}. where b t = [ n . b2 = [ il and bJ = [~] (...
 4.4.55: In Exercises 55 58. an angle 9 is given. Let v = ] and [v] 6 = [;:...
 4.4.56: In Exercises 55 58. an angle 9 is given. Let v = ] and [v] 6 = [;:...
 4.4.57: In Exercises 55 58. an angle 9 is given. Let v = ] and [v] 6 = [;:...
 4.4.58: In Exercises 55 58. an angle 9 is given. Let v = ] and [v] 6 = [;:...
 4.4.59: In Exercises 59 62. a basis B for n2 is given. If v = [;] and [v)C...
 4.4.60: In Exercises 59 62. a basis B for n2 is given. If v = [;] and [v)C...
 4.4.61: In Exercises 59 62. a basis B for n2 is given. If v = [;] and [v)C...
 4.4.62: In Exercises 59 62. a basis B for n2 is given. If v = [;] and [v)C...
 4.4.63: In Exercises 63 66. a basis B for 'R3 is given. If v = [ ~] and [ ...
 4.4.64: In Exercises 63 66. a basis B for 'R3 is given. If v = [ ~] and [ ...
 4.4.65: In Exercises 63 66. a basis B for 'R3 is given. If v = [ ~] and [ ...
 4.4.66: In Exercises 63 66. a basis B for 'R3 is given. If v = [ ~] and [ ...
 4.4.67: In Exercises 6770, an angle 0 is given. Let ,. = [;, J and [v]s = ...
 4.4.68: In Exercises 6770, an angle 0 is given. Let ,. = [;, J and [v]s = ...
 4.4.69: In Exercises 6770, an angle 0 is given. Let ,. = [;, J and [v]s = ...
 4.4.70: In Exercises 6770, an angle 0 is given. Let ,. = [;, J and [v]s = ...
 4.4.71: In Exercises 7174, a basis B for 'R2 is given. If v = [; J and [v]...
 4.4.72: In Exercises 7174, a basis B for 'R2 is given. If v = [; J and [v]...
 4.4.73: In Exercises 7174, a basis B for 'R2 is given. If v = [; J and [v]...
 4.4.74: In Exercises 7174, a basis B for 'R2 is given. If v = [; J and [v]...
 4.4.75: n Erercises 75 78, a basis B for n3 is given. ({ v = [ ~] and [v]s...
 4.4.76: n Erercises 75 78, a basis B for n3 is given. ({ v = [ ~] and [v]s...
 4.4.77: n Erercises 75 78, a basis B for n3 is given. ({ v = [ ~] and [v]s...
 4.4.78: n Erercises 75 78, a basis B for n3 is given. ({ v = [ ~] and [v]s...
 4.4.79: In Exercises 79 86. an equation of a conic section is given in the...
 4.4.80: In Exercises 79 86. an equation of a conic section is given in the...
 4.4.81: In Exercises 79 86. an equation of a conic section is given in the...
 4.4.82: In Exercises 79 86. an equation of a conic section is given in the...
 4.4.83: In Exercises 79 86. an equation of a conic section is given in the...
 4.4.84: In Exercises 79 86. an equation of a conic section is given in the...
 4.4.85: In Exercises 79 86. an equation of a conic section is given in the...
 4.4.86: In Exercises 79 86. an equation of a conic section is given in the...
 4.4.87: /n Exercises 87 94, an equation of a conic section is given in the...
 4.4.88: /n Exercises 87 94, an equation of a conic section is given in the...
 4.4.89: /n Exercises 87 94, an equation of a conic section is given in the...
 4.4.90: /n Exercises 87 94, an equation of a conic section is given in the...
 4.4.91: /n Exercises 87 94, an equation of a conic section is given in the...
 4.4.92: /n Exercises 87 94, an equation of a conic section is given in the...
 4.4.93: /n Exercises 87 94, an equation of a conic section is given in the...
 4.4.94: /n Exercises 87 94, an equation of a conic section is given in the...
 4.4.95: Let A= tu . U2 .. .. . u., l be a basis for n and c1, c2 . c. be no...
 4.4.96: Let A= {u1. u2 . .... u.,) be a basis for 'R". Recall that Exercise...
 4.4.97: Let A = ~o u2 ... , u,} be a basis for R". Recall that Exercise 73 ...
 4.4.98: Let A = ~o u2, ... , u, } be a basis for R". Recall that Exercise 7...
 4.4.99: Let A and 8 be two bases for R". If (viA = (' 'Is for some nonzero ...
 4.4.100: Let A and /3 be two bases for n. lf [v].A = [v)s for every vector v...
 4.4.101: Prove that if Sis linearly dependent, then every vector in the span...
 4.4.102: Let A and 8 be two bases for R". Express [v].A in terms of (v (6 .
 4.4.103: (a) Let 8 be a basis for R". Prove that the function T: R" + R" de...
 4.4.104: What is the standard matrix of the linear transformation T in Exerc...
 4.4.105: Let V be a subspace of R" and 8 = {u1. u2 , . Ut} be a subset of V,...
 4.4.106: Let V = { [:!] e n3 : 2v, + v2 + v3 = 0 } and (a) Show that Sis li...
 4.4.107: Let B be a basis for R" and (u 1. u2 ..... utl be a subset of R". P...
 4.4.108: Let 8 be a basis for R", (u 1, 112, .. . , uk} be a subset of R", a...
 4.4.109: Let 25 73 56 68  118 l [ 0] [ 14] [ 6] [14] [ 12] 8 = 21 ' ...
 4.4.110: For the basis /3 in Exercise I 09, lind a nonzero vector u in n5 su...
 4.4.111: For the basis 13 in Exercise 109, find a nonzero vector v in n5 suc...
 4.4.112: Lc1 Band v be as m Exerci..e 109. and lei (a) Show 1ha1 v is a line...
Solutions for Chapter 4.4: DETERMINANTS
Full solutions for Elementary Linear Algebra: A Matrix Approach  2nd Edition
ISBN: 9780131871410
Solutions for Chapter 4.4: DETERMINANTS
Get Full SolutionsChapter 4.4: DETERMINANTS includes 112 full stepbystep solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra: A Matrix Approach, edition: 2. This expansive textbook survival guide covers the following chapters and their solutions. Since 112 problems in chapter 4.4: DETERMINANTS have been answered, more than 25508 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.

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.)

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

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

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.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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.

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

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 .

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

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 .

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

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

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.