 4.6.1: Consider the bases B = {u1, u2} and B = {u 1, u 2} for R2, where u1...
 4.6.2: Repeat the directions of Exercise 1 with the same vector w but with...
 4.6.3: Consider the bases B = {u1, u2, u3} and B = {u 1, u 2, u 3} for R3,...
 4.6.4: Repeat the directions of Exercise 3 with the same vector w, but wit...
 4.6.5: Let V be the space spanned by f1 = sin x and f2 = cos x. (a) Show t...
 4.6.6: Consider the bases B = {p1, p2} and B = {q1, q2} for P1, where p1 =...
 4.6.7: Let B1 = {u1, u2} and B2 = {v1, v2} be the bases for R2 in which u1...
 4.6.8: Let S be the standard basis for R2, and let B = {v1, v2} be the bas...
 4.6.9: Let S be the standard basis for R2, and let B = {v1, v2} be the bas...
 4.6.10: Let S = {e1, e2} be the standard basis for R2, and let B = {v1, v2}...
 4.6.11: Let S = {e1, e2} be the standard basis for R2, and let B = {v1, v2}...
 4.6.12: If B1, B2, and B3 are bases for R2, and if PB1B2 = 3 1 5 2 and PB2B...
 4.6.13: If P is the transition matrix from a basis B to a basis B, and Q is...
 4.6.14: To write the coordinate vector for a vector, it is necessary to spe...
 4.6.15: Consider the matrix P = 110 102 021 (a) P is the transition matrix ...
 4.6.16: The matrix P = 100 032 011 is the transition matrix from what basis...
 4.6.17: Let S = {e1, e2} be the standard basis for R2, and let B = {v1, v2}...
 4.6.18: Let S = {e1, e2, e3} be the standard basis for R3, and let B = {v1,...
 4.6.19: If [w]B = w holds for all vectors w in Rn, what can you say about t...
 4.6.20: Let B be a basis for Rn. Prove that the vectors v1, v2,..., vk span...
 4.6.21: Let B be a basis for Rn. Prove that the vectors v1, v2,..., vk form...
Solutions for Chapter 4.6: Change of Basis
Full solutions for Elementary Linear Algebra, Binder Ready Version: Applications Version  11th Edition
ISBN: 9781118474228
Solutions for Chapter 4.6: Change of Basis
Get Full SolutionsElementary Linear Algebra, Binder Ready Version: Applications Version was written by and is associated to the ISBN: 9781118474228. This textbook survival guide was created for the textbook: Elementary Linear Algebra, Binder Ready Version: Applications Version, edition: 11. Since 21 problems in chapter 4.6: Change of Basis have been answered, more than 15577 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 4.6: Change of Basis includes 21 full stepbystep solutions.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

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.

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

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].

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Network.
A directed graph that has constants Cl, ... , Cm associated with the edges.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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.

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.