 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
ISBN: 9783319110790
Solutions for Chapter 9.A: Complexification
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra Done Right (Undergraduate Texts in Mathematics), edition: 3. Chapter 9.A: Complexification includes 19 full stepbystep solutions. Since 19 problems in chapter 9.A: Complexification have been answered, more than 5896 students have viewed full stepbystep solutions from this chapter. Linear Algebra Done Right (Undergraduate Texts in Mathematics) was written by and is associated to the ISBN: 9783319110790. This expansive textbook survival guide covers the following chapters and their solutions.

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.

Companion matrix.
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 nl  An).

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

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.

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 AI.
Square matrix with AI A = I and AAl = 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 B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Lucas numbers
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+ (MoorePenrose 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.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , 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 = MI 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.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Spanning set.
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.