 4.3.1: Explain why the following are linearly dependent sets of vectors. (...
 4.3.2: Which of the following sets of vectors in are linearly dependent? (...
 4.3.3: Which of the following sets of vectors in are linearly dependent? (...
 4.3.4: Which of the following sets of vectors in are linearly dependent? (...
 4.3.5: Assume that , , and are vectors in that have their initial points a...
 4.3.6: Assume that , , and are vectors in that have their initial points a...
 4.3.7: (a) Show that the three vectors , , and form a linearly dependent s...
 4.3.8: (a) Show that the three vectors , , and form a linearly dependent s...
 4.3.9: For which real values of do the following vectors form a linearly d...
 4.3.10: Show that if is a linearly independent set of vectors, then so are ...
 4.3.11: Show that if is a linearly independent set of vectors, then so is e...
 4.3.12: Show that if is a linearly dependent set of vectors in a vector spa...
 4.3.13: Show that if is a linearly dependent set of vectors in a vector spa...
 4.3.14: Show that in every set with more than three vectors is linearly dep...
 4.3.15: Show that if is linearly independent and does not lie in , then is ...
 4.3.16: Prove: For any vectors u, v, and w in a vector space V, the vectors...
 4.3.17: Prove: The space spanned by two vectors in is a line through the or...
 4.3.18: Under what conditions is a set with one vector linearly independent?
 4.3.19: Are the vectors , , and in part (a) of the accompanying figure line...
 4.3.20: By using appropriate identities, where required, determine which of...
 4.3.21: The functions and are linearly independent in because neither funct...
 4.3.22: The functions and are linearly independent in because neither funct...
 4.3.23: (Calculus required) Use the Wronskian to show that the following se...
 4.3.24: Show that the functions , , and are linearly independent.
 4.3.25: Show that the functions , , and are linearly independent.
 4.3.26: Use part (a) of Theorem 4.3.1 to prove part (b).
 4.3.27: Prove part (b) of Theorem 4.3.2.
 4.3.28: (a) In Example 1 we showed that the mutually orthogonal vectors i, ...
 4.3.a: In parts (a)(h) determine whether the statement is true or false, a...
 4.3.b: In parts (a)(h) determine whether the statement is true or false, a...
 4.3.c: In parts (a)(h) determine whether the statement is true or false, a...
 4.3.d: In parts (a)(h) determine whether the statement is true or false, a...
 4.3.e: In parts (a)(h) determine whether the statement is true or false, a...
 4.3.f: In parts (a)(h) determine whether the statement is true or false, a...
 4.3.g: In parts (a)(h) determine whether the statement is true or false, a...
 4.3.h: In parts (a)(h) determine whether the statement is true or false, a...
Solutions for Chapter 4.3: Linear Independence
Full solutions for Elementary Linear Algebra: Applications Version  10th Edition
ISBN: 9780470432051
Solutions for Chapter 4.3: Linear Independence
Get Full SolutionsSince 36 problems in chapter 4.3: Linear Independence have been answered, more than 14274 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Elementary Linear Algebra: Applications Version, edition: 10. Chapter 4.3: Linear Independence includes 36 full stepbystep solutions. Elementary Linear Algebra: Applications Version was written by and is associated to the ISBN: 9780470432051. This expansive textbook survival guide covers the following chapters and their solutions.

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

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!

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

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.)

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.

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

Condition number
cond(A) = c(A) = IIAIlIIAIII = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib IIĀ· Condition numbers measure the sensitivity of the output to change in the input.

Conjugate Gradient Method.
A sequence of steps (end of Chapter 9) to solve positive definite Ax = b by minimizing !x T Ax  x Tb over growing Krylov subspaces.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

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

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).

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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