- Chapter 1: Systems of Linear Equations
- Chapter 2: Matrices
- Chapter 3: Determinants
- Chapter 4: Vector Spaces
- Chapter 5: Inner Product Spaces
- Chapter 6: Inner Product Spaces
- Chapter 7: Eigenvalues and Eigenvectors
Elementary Linear Algebra 7th Edition - Solutions by Chapter
Full solutions for Elementary Linear Algebra | 7th Edition
ISBN: 9781133110873
Solutions by Chapter
Elementary Linear Algebra was written by and is associated to the ISBN: 9781133110873. The full step-by-step solution to problem in Elementary Linear Algebra were answered by , our top Math solution expert on 01/03/18, 08:36PM. Since problems from 7 chapters in Elementary Linear Algebra have been answered, more than 10495 students have viewed full step-by-step answer. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 7. This expansive textbook survival guide covers the following chapters: 7.
Key Math Terms and definitions covered in this textbook
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Affine transformation
Tv = Av + Vo = linear transformation plus shift.
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Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.
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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.
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Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).
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Complex conjugate
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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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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Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.
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Diagonalization
A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.
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Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.
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Gauss-Jordan method.
Invert A by row operations on [A I] to reach [I A-I].
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Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
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Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
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Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
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Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.
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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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Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
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Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
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Row space C (AT) = all combinations of rows of A.
Column vectors by convention.