 4.5.1: Label the following statements as true or false. (a) Any nlinear f...
 4.5.2: Determine all the 1linear functions 8: M) x i(F) F
 4.5.3: Determine which of the functions 8: M3X.}(F) * F in Exercises 310 ...
 4.5.4: Determine which of the functions 8: M3X.}(F) * F in Exercises 310 ...
 4.5.5: Determine which of the functions 8: M3X.}(F) * F in Exercises 310 ...
 4.5.6: Determine which of the functions 8: M3X.}(F) * F in Exercises 310 ...
 4.5.7: Determine which of the functions 8: M3X.}(F) * F in Exercises 310 ...
 4.5.8: Determine which of the functions 8: M3X.}(F) * F in Exercises 310 ...
 4.5.9: Determine which of the functions 8: M3X.}(F) * F in Exercises 310 ...
 4.5.10: Determine which of the functions 8: M3X.}(F) * F in Exercises 310 ...
 4.5.11: Prove Corollaries 2 and 3 of Theorem 4.10.
 4.5.12: Prove Theorem 4.11.
 4.5.13: Prove that det: M2x2 (F) > F is a 2linear function of the columns ...
 4.5.14: Let a,b,c,d G F. Prove that the function 8: M2X2(F) * F defined by ...
 4.5.15: Prove that 8: M2x2 (F) F is a 2linear function if and only if it h...
 4.5.16: Prove that if 8: MnXT1(F) > F is an alternating nlinear function, ...
 4.5.17: Prove that a linear combination of two nlinear functions is an nl...
 4.5.18: Prove that the set of all nlinear functions over a field F is a ve...
 4.5.19: Let 8: MnXn (F) * F be an nlinear function and F a field that does...
 4.5.20: Give an example to show that the implication in Exercise 19 need no...
Solutions for Chapter 4.5: A Characterization of the Determinant
Full solutions for Linear Algebra  4th Edition
ISBN: 9780130084514
Solutions for Chapter 4.5: A Characterization of the Determinant
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 4.5: A Characterization of the Determinant includes 20 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra , edition: 4. Linear Algebra was written by and is associated to the ISBN: 9780130084514. Since 20 problems in chapter 4.5: A Characterization of the Determinant have been answered, more than 10185 students have viewed full stepbystep solutions from this chapter.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Block matrix.
A matrix can be partitioned into matrix blocks, by cuts between rows and/or between columns. Block multiplication ofAB is allowed if the block shapes permit.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

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.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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!).

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Iterative method.
A sequence of steps intended to approach the desired solution.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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

Special solutions to As = O.
One free variable is Si = 1, other free variables = o.