 4.4.1: Use the method of Example 3 to show that the following set of vecto...
 4.4.2: Use the method of Example 3 to show that the following set of vecto...
 4.4.3: Show that the following polynomials form a basis for P2. x2 + 1, x2...
 4.4.4: Show that the following polynomials form a basis for P3. 1 + x, 1 x...
 4.4.5: Show that the following matrices form a basis for M22. 3 6 3 6 , 0 ...
 4.4.6: Show that the following matrices form a basis for M22. 1 1 1 1 , 1 ...
 4.4.7: In each part, show that the set of vectors is not a basis for R3. (...
 4.4.8: Show that the following vectors do not form a basis for P2. 1 3x + ...
 4.4.9: Show that the following matrices do not form a basis for M22. 1 0 1...
 4.4.10: Let V be the space spanned by v1 = cos2 x, v2 = sin2 x, v3 = cos 2x...
 4.4.11: Find the coordinate vector of w relative to the basis S = {u1, u2} ...
 4.4.12: Find the coordinate vector of w relative to the basis S = {u1, u2} ...
 4.4.13: Find the coordinate vector of v relative to the basis S = {v1, v2, ...
 4.4.14: Find the coordinate vector of v relative to the basis S = {v1, v2, ...
 4.4.15: In Exercises 1516, first show that the set S = {A1, A2, A3, A4} is ...
 4.4.16: In Exercises 1516, first show that the set S = {A1, A2, A3, A4} is ...
 4.4.17: In Exercises 1718, first show that the set S = {p1, p2, p3} is a ba...
 4.4.18: In Exercises 1718, first show that the set S = {p1, p2, p3} is a ba...
 4.4.19: In words, explain why the sets of vectors in parts (a) to (d) are n...
 4.4.20: In any vector space a set that contains the zero vector must be lin...
 4.4.21: In each part, let TA: R3 R3 be multiplication by A, and let {e1, e2...
 4.4.22: In each part, let TA: R3 R3 be multiplication by A, and let u = (1,...
 4.4.23: The accompanying figure shows a rectangular xycoordinate system de...
 4.4.24: The accompanying figure shows a rectangular xycoordinate system an...
 4.4.25: The first four Hermite polynomials [named for the French mathematic...
 4.4.26: The first four Hermite polynomials [named for the French mathematic...
 4.4.27: Consider the coordinate vectors [w]S = 6 1 4 , [q]S = 3 0 4 , [B]S ...
 4.4.28: The basis that we gave for M22 in Example 4 consisted of noninverti...
 4.4.29: Prove that R is an infinitedimensional vector space.
 4.4.30: Let TA: Rn Rn be multiplication by an invertible matrix A, and let ...
 4.4.31: Prove that if V is a subspace of a vector space W and if V is infin...
Solutions for Chapter 4.4: Coordinates and Basis
Full solutions for Elementary Linear Algebra, Binder Ready Version: Applications Version  11th Edition
ISBN: 9781118474228
Solutions for Chapter 4.4: Coordinates and Basis
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Since 31 problems in chapter 4.4: Coordinates and Basis have been answered, more than 17205 students have viewed full stepbystep solutions from this chapter. Elementary 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. Chapter 4.4: Coordinates and Basis includes 31 full stepbystep solutions.

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.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

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.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

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].

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Minimal polynomial of A.
The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A  AI) if no eigenvalues are repeated; always meA) divides peA).

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

Normal matrix.
If N NT = NT N, then N has orthonormal (complex) eigenvectors.

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.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Reflection matrix (Householder) Q = I 2uuT.
Unit vector u is reflected to Qu = u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q1 = Q.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn 1 with P(Xi) = bi. Vij = (Xi)jI and det V = product of (Xk  Xi) for k > i.