- 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 8th Edition - Solutions by Chapter
Full solutions for Linear Algebra with Applications | 8th Edition
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.)
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).
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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].
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.
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).
Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + Fn- 2 = (A7 -A~)I()q -A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].
Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.
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.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
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.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
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.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
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.
Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •
Outer product uv T
= column times row = rank one matrix.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.