- 9.A.1: Prove 9.3.
- 9.A.2: Verify that if V is a real vector space and T 2 L.V /, then TC 2 L....
- 9.A.3: Suppose V is a real vector space and v1;:::; vm 2 V. Prove thatv1;:...
- 9.A.4: Suppose V is a real vector space and v1;:::; vm 2 V. Prove thatv1;:...
- 9.A.5: Suppose that V is a real vector space and S;T 2 L.V /. Show that.S ...
- 9.A.6: Suppose V is a real vector space and T 2 L.V /. Prove that TC isinv...
- 9.A.7: Suppose V is a real vector space and N 2 L.V /. Prove that NC isnil...
- 9.A.8: Suppose T 2 L.R3/ and 5; 7 are eigenvalues of T. Prove that TC has ...
- 9.A.9: Prove there does not exist an operator T 2 L.R7/ such that T 2 C T ...
- 9.A.10: Give an example of an operator T 2 L.C7/ such that T 2 C T C Iis ni...
- 9.A.11: Suppose V is a real vector space and T 2 L.V /. Suppose there exist...
- 9.A.12: Suppose V is a real vector space and T 2 L.V /. Suppose there exist...
- 9.A.13: Suppose V is a real vector space, T 2 L.V /, and b; c 2 R are such ...
- 9.A.14: Suppose V is a real vector space with dim V D 8. Suppose T 2 L.V /i...
- 9.A.15: Suppose V is a real vector space and T 2 L.V / has no eigenvalues.P...
- 9.A.16: Suppose V is a real vector space. Prove that there exists T 2 L.V /...
- 9.A.17: Suppose V is a real vector space and T 2 L.V / satisfies T 2 D I.De...
- 9.A.18: Suppose V is a real vector space and T 2 L.V /. Prove that the foll...
- 9.A.19: Suppose V is a real vector space with dim V D n and T 2 L.V / issuc...
Solutions for Chapter 9.A: Complexification
Full solutions for Linear Algebra Done Right (Undergraduate Texts in Mathematics) | 3rd Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.
Put CI, ... ,Cn in row n and put n - 1 ones just above the main diagonal. Then det(A - AI) = ±(CI + c2A + C3A 2 + .•. + cnA n-l - An).
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.
0,1,1,2,3,5, ... satisfy Fn = Fn-l + 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.
Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.
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.
Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or -1) based on the number of row exchanges to reach I.
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).
Rank r (A)
= number of pivots = dimension of column space = dimension of row space.
R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.
Similar matrices A and B.
Every B = M-I AM has the same eigenvalues as 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.
Skew-symmetric matrix K.
The transpose is -K, since Kij = -Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.