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College Algebra and Trigonometry 7th Edition - Solutions by Chapter
Full solutions for College Algebra and Trigonometry | 7th Edition
ISBN: 9781439048603
Solutions by Chapter
This textbook survival guide was created for the textbook: College Algebra and Trigonometry, edition: 7. The full step-by-step solution to problem in College Algebra and Trigonometry were answered by , our top Math solution expert on 11/15/17, 04:29PM. This expansive textbook survival guide covers the following chapters: 12. College Algebra and Trigonometry was written by and is associated to the ISBN: 9781439048603. Since problems from 12 chapters in College Algebra and Trigonometry have been answered, more than 20893 students have viewed full step-by-step answer.
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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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].
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Gram-Schmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.
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Iterative method.
A sequence of steps intended to approach the desired solution.
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Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
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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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Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.
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Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.
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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.
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Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
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Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
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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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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.
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Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
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Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
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Schur complement S, D - C A -} B.
Appears in block elimination on [~ g ].
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Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.
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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.
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Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
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Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.