- 1.B.1: Prove that .v/ D v for every v 2 V.
- 1.B.2: Suppose a 2 F, v 2 V, and av D 0. Prove that a D 0 or v D 0.
- 1.B.3: Suppose v;w 2 V. Explain why there exists a unique x 2 V such thatv...
- 1.B.4: The empty set is not a vector space. The empty set fails to satisfy...
- 1.B.5: Show that in the definition of a vector space (1.19), the additive ...
- 1.B.6: Let 1 and 1 denote two distinct objects, neither of which is in R.D...
Solutions for Chapter 1.B: Definition of Vector Space
Full solutions for Linear Algebra Done Right (Undergraduate Texts in Mathematics) | 3rd Edition
Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.
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.
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].
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.
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).
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Free columns of A.
Columns without pivots; these are combinations of earlier columns.
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.
Left inverse A+.
If A has full column rank n, then A+ = (AT A)-I AT has A+ A = In.
Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).
Every v in V is orthogonal to every w in W.
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).
Singular matrix A.
A square matrix that has no inverse: det(A) = o.
Constant down each diagonal = time-invariant (shift-invariant) filter.
Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·
Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.