 4.5.1: Show that
 4.5.2: Show that is a linearly independent sel in R3.
 4.5.3: Determine whether IS a linearly independem set in R4
 4.5.4: Determine whether s = l[l ,].[3 8  5].[3 6  9]) IS a linearly in...
 4.5.5: III E.terc:il/,S 5 throllgh 8, "ch gil'(II /lfIgmellled IIllIlrix i...
 4.5.6: III E.terc:il/,S 5 throllgh 8, "ch gil'(II /lfIgmellled IIllIlrix i...
 4.5.7: III E.terc:il/,S 5 throllgh 8, "ch gil'(II /lfIgmellled IIllIlrix i...
 4.5.8: III E.terc:il/,S 5 throllgh 8, "ch gil'(II /lfIgmellled IIllIlrix i...
 4.5.9: L" " = [il" = [=} , = m ,"g
 4.5.10: L" " = [H' , = [t}" = m relo"g
 4.5.11: Which o f the given vectors in RJ are linearly JepenJent? For those...
 4.5.12: Consider the vector space ~:. Follow the directions of Exercise II....
 4.5.13: Consider the vector space 1'2. Follow the directims o f Exercise II...
 4.5.14: Let V be the vector space of all realvalued continuous function s....
 4.5.15: Consider the vector nce RJ. Fo llow the directions of Exercise II. ...
 4.5.16: For what values of c are the veCIOl'S [I 0  I J. 12 2], and [ I c...
 4.5.17: For wh:1I vn lu e~ of (' [.re the veCIOl'S t + 3 and 21 + c1 + 2 in...
 4.5.18: Lei u and v be nonzero veCIOrs in a vector ~pace V. Show thai u and...
 4.5.19: Let S = {VI. vl .' ,. \'.1 1 be a SCI of vectors in a vector space ...
 4.5.20: Suppose that S = l VI. v!, vJ! is a line3r1y ind
 4.5.21: Suppose that S = (VI. '0' 2. '0'3) is a linearly independent set of...
 4.5.22: uppose that S = {VI. '0'2. \ ') 1 is a linearly dependent set of ve...
 4.5.23: Show that if (VI. V2) is lineally independelll and v] does aot belo...
 4.5.24: Suppose that {VI. '0'2 ..... "N} is a linearly independent set of v...
 4.5.25: Let A be an //I x /I matrix in reduced row echelon form. Prove that...
 4.5.26: Let S = {UI. U2 ..... UI'} be a set of vectors in a vector space an...
 4.5.27: Let Sj and S2 be finite subsets of a vector space and let 51 be a s...
 4.5.28: Let SI and S2 be finite subsets of a vector space and let 51 be a s...
 4.5.29: Let A be an 1/1 x /I matrix. Associate with A the vector win RN '" ...
 4.5.30: As noted in the Remark after Example 7 in Section 4.4. to detennine...
 4.5.31: (\Vurning: TIle stratcgy given in Exercisc 30 assumes the computati...
Solutions for Chapter 4.5: Linear Independence
Full solutions for Elementary Linear Algebra with Applications  9th Edition
ISBN: 9780132296540
Solutions for Chapter 4.5: Linear Independence
Get Full SolutionsThis textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9. Chapter 4.5: Linear Independence includes 31 full stepbystep solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 31 problems in chapter 4.5: Linear Independence have been answered, more than 13444 students have viewed full stepbystep solutions from this chapter. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780132296540.

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.

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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.

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

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

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.

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.

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

Incidence matrix of a directed graph.
The m by n edgenode incidence matrix has a row for each edge (node i to node j), with entries 1 and 1 in columns i and j .

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).

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).

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

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

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.

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

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)!

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.