 5.1.1: In Exercises 14, find the coordinate vector of u relative to the g...
 5.1.2: In Exercises 14, find the coordinate vector of u relative to the g...
 5.1.3: In Exercises 14, find the coordinate vector of u relative to the g...
 5.1.4: In Exercises 14, find the coordinate vector of u relative to the g...
 5.1.5: In Exercises 58, find the coordinate vector of u relative to the g...
 5.1.6: In Exercises 58, find the coordinate vector of u relative to the g...
 5.1.7: In Exercises 58, find the coordinate vector of u relative to the g...
 5.1.8: In Exercises 58, find the coordinate vector of u relative to the g...
 5.1.9: In Exercises 912, find the coordinate vector ofu relative to the g...
 5.1.10: In Exercises 912, find the coordinate vector ofu relative to the g...
 5.1.11: In Exercises 912, find the coordinate vector ofu relative to the g...
 5.1.12: In Exercises 912, find the coordinate vector ofu relative to the g...
 5.1.13: In Exercises 1315, find the coordinate vector of u relative to the...
 5.1.14: In Exercises 1315, find the coordinate vector of u relative to the...
 5.1.15: In Exercises 1315, find the coordinate vector of u relative to the...
 5.1.16: In Exercises 1620, find the transition matrix P from the given ba...
 5.1.17: In Exercises 1620, find the transition matrix P from the given ba...
 5.1.18: In Exercises 1620, find the transition matrix P from the given ba...
 5.1.19: In Exercises 1620, find the transition matrix P from the given ba...
 5.1.20: In Exercises 1620, find the transition matrix P from the given ba...
 5.1.21: Consider the bases B = {{1, 0), {O, 1)} and B' = {{5, 3), (3, 2)} o...
 5.1.22: Consider the bases B = {{l, 0), (0, 1)} and B' = {{1, 2), (1, 1)}...
 5.1.23: Find the transition matrix P from the basis B = {(l, 2), (3, O)} of...
 5.1.24: Find the transition matrix P from the basis B = {{3, 1), (1, l )} ...
 5.1.25: Consider the vector space P 2 B = {x2, x, 1} and B' = {3x2, x  1, ...
 5.1.26: Consider the vector space P1 B = {x, 1} and B' = { x + 2, 3} are ba...
 5.1.27: Construct an isomorphism from the vector space of diagonal 2 X 2 ma...
 5.1.28: Construct an isomorphism from the vector space of symmetric 2 X 2 m...
 5.1.29: Let Tbe a matrix transformation defined by a square matrix A. Prove...
 5.1.30: Let A be a nonsingular matrix that defines an isomorphism of Rn ont...
Solutions for Chapter 5.1: Coordinate Vectors
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9781449679545
Solutions for Chapter 5.1: Coordinate Vectors
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 8. Since 30 problems in chapter 5.1: Coordinate Vectors have been answered, more than 8663 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9781449679545. Chapter 5.1: Coordinate Vectors includes 30 full stepbystep solutions.

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

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

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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.

Circulant matrix C.
Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn  l . Cx = convolution c * x. Eigenvectors in F.

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).

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Covariance matrix:E.
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.

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].

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.

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).