 1.7.1E: In Exercises 1–4, determine if the vectors are linearly independent...
 1.7.2E: Determine if the vectors are linearly independent. Justify each ans...
 1.7.3E: Determine if the vectors are linearly independent. Justify each ans...
 1.7.4E: Determine if the vectors are linearly independent. Justify each ans...
 1.7.5E: Determine if the columns of the matrix form a linearly independent ...
 1.7.6E: Determine if the columns of the matrix form a linearly independent ...
 1.7.7E: In Exercises 5–8, determine if the columns of the matrix form a lin...
 1.7.8E: Determine if the columns of the matrix form a linearly independent ...
 1.7.9E: In Exercises 9 and 10, (a) for what values of h is v3 in Span {v1, ...
 1.7.10E: (a) for what values of h is v3 in Span {v1, v2}, and (b) for what v...
 1.7.11E: Find the value(s) of h for which the vectors are linearly dependent...
 1.7.12E: Find the value(s) of h for which the vectors are linearly dependent...
 1.7.13E: In Exercises 11–14, find the value(s) of h for which the vectors ar...
 1.7.14E: Find the value(s) of h for which the vectors are linearly dependent...
 1.7.15E: Determine by inspection whether the vectors in Exercises 15–20 are ...
 1.7.16E: Determine by inspection whether the vectors in Exercises are linear...
 1.7.17E: Determine by inspection whether the vectors in Exercises are linear...
 1.7.18E: Determine by inspection whether the vectors in Exercises are linear...
 1.7.19E: Determine by inspection whether the vectors in Exercises 15–20 are ...
 1.7.20E: Determine by inspection whether the vectors in Exercises 15–20 are ...
 1.7.21E: In Exercises 21 and 22, mark each statement True or False. Justify ...
 1.7.22E: a. Two vectors are linearly dependent if and only if they lie on a ...
 1.7.23E: In Exercises 23–26, describe the possible echelon forms of the matr...
 1.7.24E: In Exercises 23–26, describe the possible echelon forms of the matr...
 1.7.25E: In Exercises 23–26, describe the possible echelon forms of the matr...
 1.7.26E: In Exercises 23–26, describe the possible echelon forms of the matr...
 1.7.27E: How many pivot columns must a 7 × 5 matrix have if its columns are ...
 1.7.28E: How many pivot columns must a 5 × 7 matrix have if its columns span...
 1.7.29E: Construct 3 × 2 matrices A and B such that Ax = 0 has a nontrivial ...
 1.7.30E: a. Fill in the blank in the following statement: “If A is an m × n ...
 1.7.31E: Exercises 31 and 32 should be solved without performing row operati...
 1.7.32E: Given A = observe that the first column plus twice the second colum...
 1.7.33E: Each statement in Exercises 33–38 is either true (in all cases) or ...
 1.7.34E: If v1,...,v4 are in ?4 and v3 =0, then {v1, v2, v3, v4} is linearly...
 1.7.35E: Each statement in Exercises 33–38 is either true (in all cases) or ...
 1.7.36E: If v1,..., v4 are in ?4 and v3 is not a linear combination of v1, v...
 1.7.37E: Each statement in Exercises 33–38 is either true (in all cases) or ...
 1.7.38E: Each statement in Exercises 33–38 is either true (in all cases) or ...
 1.7.39E: Suppose A is an m × n matrix with the property that for all b in ?m...
 1.7.40E: Suppose an m × n matrix A has n pivot columns. Explain why for each...
 1.7.41E: Use as many columns of A as possible to construct a matrix B with t...
 1.7.42E: Use as many columns of A as possible to construct a matrix B with t...
 1.7.43E: [M] With A and B as in Exercise 41, select a column v of A that was...
 1.7.44E: [M] Repeat Exercise 43 with the matrices A and B from Exercise 42. ...
Solutions for Chapter 1.7: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 1.7
Get Full SolutionsChapter 1.7 includes 44 full stepbystep solutions. This textbook survival guide was created for the textbook: Linear Algebra and Its Applications , edition: 5th. Since 44 problems in chapter 1.7 have been answered, more than 11425 students have viewed full stepbystep solutions from this chapter. Linear Algebra and Its Applications was written by Sieva Kozinsky and is associated to the ISBN: 9780321982384. This expansive textbook survival guide covers the following chapters and their solutions.

Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

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.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

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.

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

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.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
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