- 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
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.)
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Invert A by row operations on [A I] to reach [I A-I].
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
A symmetric matrix with eigenvalues of both signs (+ and - ).
Jordan form 1 = M- 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.
Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
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 •
Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.
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.
Row space C (AT) = all combinations of rows of A.
Column vectors by convention.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and A-I are BT AT and (AT)-I.