 3.5.1: For each of the following, find the transition matrix corresponding...
 3.5.2: For each of the ordered bases {u1, u2} in Exercise 1, find the tran...
 3.5.3: Let v1 = (3, 2)T and v2 = (4, 3)T . For each ordered basis {u1, u2}...
 3.5.4: Let E = [(5, 3)T , (3, 2)T ] and let x = (1, 1)T , y = (1,1)T , and...
 3.5.5: Let u1 = (1, 1, 1)T , u2 = (1, 2, 2)T , u3 = (2, 3, 4)T . (a) Find ...
 3.5.6: Let v1 = (4, 6, 7)T , v2 = (0, 1, 1)T , v3 = (0, 1, 2)T , and let u...
 3.5.7: Given v1 = 1 2 , v2 = 2 3 , S = 3 5 1 2 find vectors w1 and w2 so t...
 3.5.8: Given v1 = 2 6 , v2 = 1 4 , S = 4 1 2 1 find vectors u1 and u2 so t...
 3.5.9: Let [x, 1] and [2x 1, 2x +1] be ordered bases for P2. (a) Find the ...
 3.5.10: Find the transition matrix representing the change of coordinates o...
 3.5.11: Let E = {u1, . . . , un} and F = {v1, . . . , vn} be two ordered ba...
Solutions for Chapter 3.5: Change of Basis
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9780136009290
Solutions for Chapter 3.5: Change of Basis
Get Full SolutionsChapter 3.5: Change of Basis includes 11 full stepbystep solutions. Since 11 problems in chapter 3.5: Change of Basis have been answered, more than 5070 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Linear Algebra with Applications was written by and is associated to the ISBN: 9780136009290.

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.

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

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.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Outer product uv T
= column times row = rank one matrix.

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

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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

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.

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

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.