- Chapter 1: Matrices and Systems of Equations
- Chapter 1.1: Systems of Linear Equations
- Chapter 1.2: Row Echelon Form
- Chapter 1.3: Matrix Arithmetic
- Chapter 1.4: Matrix Algebra
- Chapter 1.5: Elementary Matrices
- Chapter 1.6: Partitioned Matrices
- Chapter 2: Determinants
- Chapter 2.1: The Determinant of a Matrix
- Chapter 2.2: Properties of Determinants
- Chapter 2.3: Additional Topics and Applications
- Chapter 3: Vector Spaces
- Chapter 3.1: Definition and Examples
- Chapter 3.2: Subspaces
- Chapter 3.3: Linear Independence
- Chapter 3.4: Basis and Dimension
- Chapter 3.5: Change of Basis
- Chapter 3.6: Row Space and Column Space
- Chapter 4: Linear Transformations
- Chapter 4.1: Definition and Examples
- Chapter 4.2: Matrix Representations of Linear Transformations
- Chapter 4.3: Similarity
- Chapter 5: Orthogonality
- Chapter 5.1: The Scalar Product in Rn
- Chapter 5.2: Orthogonal Subspaces
- Chapter 5.3: Least Squares Problems
- Chapter 5.4: Inner Product Spaces
- Chapter 5.5: Orthonormal Sets
- Chapter 5.6: The GramSchmidt Orthogonalization Process
- Chapter 5.7: Orthogonal Polynomials
- Chapter 6: Eigenvalues
- Chapter 6.1: Eigenvalues and Eigenvectors
- Chapter 6.2: Systems of Linear Differential Equations
- Chapter 6.3: Diagonalization
- Chapter 6.4: Hermitian Matrices
- Chapter 6.5: The Singular Value Decomposition
- Chapter 6.6: Quadratic Forms
- Chapter 6.7: Positive Definite Matrices
- Chapter 6.8: Nonnegative Matrices
- Chapter 7: Numerical Linear Algebra
- Chapter 7.1: Floating-Point Numbers
- Chapter 7.2: Gaussian Elimination
- Chapter 7.3: Pivoting Strategies
- Chapter 7.4: Matrix Norms and Condition Numbers
- Chapter 7.5: Orthogonal Transformations
- Chapter 7.6: The Eigenvalue Problem
- Chapter 7.7: Least Squares Problems
Linear Algebra with Applications 9th Edition - Solutions by Chapter
Full solutions for Linear Algebra with Applications | 9th Edition
ISBN: 9780321962218
Linear Algebra with Applications | 9th Edition - Solutions by Chapter
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Solutions by Chapter
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Key Math Terms and definitions covered in this textbook
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Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!
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Cayley-Hamilton Theorem.
peA) = det(A - AI) has peA) = zero matrix.
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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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Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
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Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A
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Dimension of vector space
dim(V) = number of vectors in any basis for V.
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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).
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Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.
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Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.
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Gauss-Jordan method.
Invert A by row operations on [A I] to reach [I A-I].
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Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
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Identity matrix I (or In).
Diagonal entries = 1, off-diagonal entries = 0.
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Matrix multiplication AB.
The i, j entry of AB is (row i of A)ยท(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 .
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Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.
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Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.
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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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Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
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Solvable system Ax = b.
The right side b is in the column space of A.
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Special solutions to As = O.
One free variable is Si = 1, other free variables = o.
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Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).