 3.3.1: Determine whether the following vectors are linearly independent in...
 3.3.2: Determine whether the following vectors are linearly independent in...
 3.3.3: For each of the sets of vectors in Exercise 2, describe geometrical...
 3.3.4: Determine whether the following vectors are linearly independent in...
 3.3.5: Let x1, x2, . . . , xk be linearly independent vectors in a vector ...
 3.3.6: Let x1, x2, and x3 be linearly independent vectors in Rn and let y1...
 3.3.7: Let x1, x2, and x3 be linearly independent vectors in Rn and let y1...
 3.3.8: Determine whether the following vectors are linearly independent in...
 3.3.9: For each of the following, show that the given vectors are linearly...
 3.3.10: Determine whether the vectors cos x, 1, sin2(x/2) are linearly inde...
 3.3.11: Consider the vectors cos(x + ) and sin x in C[, ]. For what values ...
 3.3.12: Given the functions 2x and x, show that (a) these two vectors are...
 3.3.13: Prove that any finite set of vectors that contains the zero vector ...
 3.3.14: Let v1 and v2 be two vectors in a vector space V. Show that v1 and ...
 3.3.15: Prove that any nonempty subset of a linearly independent set of vec...
 3.3.16: Let A be an m n matrix. Show that if A has linearly independent col...
 3.3.17: Let x1, . . . , xk be linearly independent vectors in Rn, and let A...
 3.3.18: Let A be a 3 3 matrix and let x1, x2, and x3 be vectors in R3. Show...
 3.3.19: Let {v1, . . . , vn} be a spanning set for the vector space V, and ...
 3.3.20: Let v1, v2, . . . , vn be linearly independent vectors in a vector ...
Solutions for Chapter 3.3: Linear Independence
Full solutions for Linear Algebra with Applications  8th Edition
ISBN: 9780136009290
Solutions for Chapter 3.3: Linear Independence
Get Full SolutionsLinear Algebra with Applications was written by and is associated to the ISBN: 9780136009290. Chapter 3.3: Linear Independence includes 20 full stepbystep solutions. Since 20 problems in chapter 3.3: Linear Independence have been answered, more than 7453 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.

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.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

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.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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

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.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

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

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(DÂ» O.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

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

Solvable system Ax = b.
The right side b is in the column space of A.