- 3.C.1: Suppose V and W are finite-dimensional and T 2 L.V; W /. Show thatw...
- 3.C.2: Suppose D 2 LP3.R/;P2.R/is the differentiation map defined byDp D p...
- 3.C.3: Suppose V and W are finite-dimensional and T 2 L.V; W /. Provethat ...
- 3.C.4: Suppose v1;:::; vm is a basis of V and W is finite-dimensional. Sup...
- 3.C.5: Suppose v1;:::; vm is a basis of V and W is finite-dimensional. Sup...
- 3.C.6: Suppose V and W are finite-dimensional and T 2 L.V; W /. Prove that...
- 3.C.7: Verify 3.36
- 3.C.8: Verify 3.38
- 3.C.9: Prove 3.52
- 3.C.10: Suppose A is an m-by-n matrix and C is an n-by-p matrix. Prove that...
- 3.C.11: Suppose a D a1 anis a 1-by-n matrix and C is an n-by-pmatrix. Prove...
- 3.C.12: Give an example with 2-by-2 matrices to show that matrix multiplica...
- 3.C.13: Prove that the distributive property holds for matrix addition and ...
- 3.C.14: Prove that matrix multiplication is associative. In other words, su...
- 3.C.15: Suppose A is an n-by-n matrix and 1 j; k n. Show that the entry inr...
Solutions for Chapter 3.C: Matrices
Full solutions for Linear Algebra Done Right (Undergraduate Texts in Mathematics) | 3rd Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
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!
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).
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x - x) (x - x) T is positive (semi)definite; :E is diagonal if the Xi are independent.
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
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.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.
Incidence matrix of a directed graph.
The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .
A symmetric matrix with eigenvalues of both signs (+ and - ).
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.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
= Xl (column 1) + ... + xn(column n) = combination of columns.
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 •
Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.
Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
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.