- 4.5.1: Label the following statements as true or false. (a) Any n-linear f...
- 4.5.2: Determine all the 1-linear functions 8: M) x i(F) F
- 4.5.3: Determine which of the functions 8: M3X.}(F) * F in Exercises 3-10 ...
- 4.5.4: Determine which of the functions 8: M3X.}(F) * F in Exercises 3-10 ...
- 4.5.5: Determine which of the functions 8: M3X.}(F) * F in Exercises 3-10 ...
- 4.5.6: Determine which of the functions 8: M3X.}(F) * F in Exercises 3-10 ...
- 4.5.7: Determine which of the functions 8: M3X.}(F) * F in Exercises 3-10 ...
- 4.5.8: Determine which of the functions 8: M3X.}(F) * F in Exercises 3-10 ...
- 4.5.9: Determine which of the functions 8: M3X.}(F) * F in Exercises 3-10 ...
- 4.5.10: Determine which of the functions 8: M3X.}(F) * F in Exercises 3-10 ...
- 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 2-linear 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 2-linear function if and only if it h...
- 4.5.16: Prove that if 8: MnXT1(F) > F is an alternating n-linear function, ...
- 4.5.17: Prove that a linear combination of two n-linear functions is an n-l...
- 4.5.18: Prove that the set of all n-linear functions over a field F is a ve...
- 4.5.19: Let 8: MnXn (F) * F be an n-linear 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
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Upper triangular systems are solved in reverse order Xn to Xl.
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.
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
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.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Inverse matrix A-I.
Square matrix with A-I A = I and AA-l = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.
Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
Reflection matrix (Householder) Q = I -2uuT.
Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.
Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.