 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: FloatingPoint 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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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!

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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.)

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Dimension of vector space
dim(V) = number of vectors in any basis for V.

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).

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.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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 .

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

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).

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).