 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: 4th. 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 5171 students have viewed full stepbystep solutions from this chapter.

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.

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column space C (A) =
space of all combinations of the columns of A.

Cramer's Rule for Ax = b.
B j has b replacing column j of A; x j = det B j I det A

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).
I don't want to reset my password
Need help? Contact support
Having trouble accessing your account? Let us help you, contact support at +1(510) 9441054 or support@studysoup.com
Forgot password? Reset it here