- Chapter 1: Vectors
- Chapter 2: Systems of Linear Equations
- Chapter 3: Matrices
- Chapter 4: Eigenvalues and Eigenvectors
- Chapter 5: Orhthogonality
- Chapter 6: Vector Spaces
- Chapter 7: Vector Spaces
Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) 3rd Edition - Solutions by Chapter
Full solutions for Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) | 3rd Edition
ISBN: 9780538735452
Linear Algebra: A Modern Introduction (Available 2011 Titles Enhanced Web Assign) | 3rd Edition - Solutions by Chapter
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Key Math Terms and definitions covered in this textbook
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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.)
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Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax - x Tb over growing Krylov subspaces.
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Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.
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lA-II = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n - 1, volume of box = I det( A) I.
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Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
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Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
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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.
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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).
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Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.
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Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
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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).
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Rotation matrix
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
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Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
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Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
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Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).
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Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.
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Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.
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Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
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Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).