 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
ISBN: 9783319110790
Solutions for Chapter 1.B: Definition of Vector Space
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra Done Right (Undergraduate Texts in Mathematics), edition: 3. Linear Algebra Done Right (Undergraduate Texts in Mathematics) was written by and is associated to the ISBN: 9783319110790. Chapter 1.B: Definition of Vector Space includes 6 full stepbystep solutions. Since 6 problems in chapter 1.B: Definition of Vector Space have been answered, more than 5789 students have viewed full stepbystep solutions from this chapter.

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.

Covariance matrix:E.
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.

Distributive Law
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 edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

Indefinite matrix.
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).

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

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).

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) 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)jI and det V = product of (Xk  Xi) for k > i.