 2.3.1E: Unless otherwise specified, assume that all matrices in these exerc...
 2.3.2E: Unless otherwise specified, assume that all matrices in these exerc...
 2.3.3E: Unless otherwise specified, assume that all matrices in these exerc...
 2.3.4E: Unless otherwise specified, assume that all matrices in these exerc...
 2.3.5E: Unless otherwise specified, assume that all matrices in these exerc...
 2.3.6E: Unless otherwise specified, assume that all matrices in these exerc...
 2.3.7E: Unless otherwise specified, assume that all matrices in these exerc...
 2.3.8E: Unless otherwise specified, assume that all matrices in these exerc...
 2.3.9E: Unless otherwise specified, assume that all matrices in these exerc...
 2.3.10E: Unless otherwise specified, assume that all matrices in these exerc...
 2.3.11E: In Exercises 11 and 12, the matrices are all n × n. Each part of th...
 2.3.12E: In Exercises 11 and 12, the matrices are all n × n. Each part of th...
 2.3.13E: An m × n upper triangular matrix is one whose entries below the mai...
 2.3.14E: An m × n lower triangular matrix is one whose entries below the mai...
 2.3.15E: Is it possible for a 4 × 4 matrix to be invertible when its columns...
 2.3.16E: If an n × n matrix A is invertible, then the columns of AT are line...
 2.3.17E: Can a square matrix with two identical columns be invertible? Why o...
 2.3.18E: Can a square matrix with two identical rows be invertible? Why or w...
 2.3.19E: If the columns of a 7 × 7 matrix D are linearly independent, what c...
 2.3.20E: If A is a 5 × 5 matrix and the equation Ax = b is consistent for ev...
 2.3.21E: If the equation Cu = v has more than one solution for some v in ?n,...
 2.3.22E: If n × n matrices E and F have the property that EF = I, then E and...
 2.3.23E: Assume that F is an n × n matrix. If the equation Fx = y is inconsi...
 2.3.24E: If an n × n matrix G cannot be row reduced to In, what can you say ...
 2.3.25E: Verify the boxed statement preceding Example 1.Example 1:Use the In...
 2.3.26E: Explain why the columns of A2 span ?n whenever the columns of an n ...
 2.3.27E: Let A and B be n × n matrices. Show that if AB is invertible, so is...
 2.3.28E: Let A and B be n × n matrices. Show that if AB is invertible, so is B.
 2.3.29E: If A is an n × n matrix and the transformation x ? Ax is onetoone...
 2.3.30E: If A is an n × n matrix and the equation Ax = b has more than one s...
 2.3.31E: Suppose A is an n × n matrix with the property that the equation Ax...
 2.3.32E: Suppose A is an n × n matrix with the property that the equation Ax...
 2.3.33E: In Exercises 33 and 34, T is a linear transformation from ?2 into ?...
 2.3.34E: In Exercises 33 and 34, T is a linear transformation from ?3 into ?...
 2.3.35E: Let T : ?n? ?n be an invertible linear transformation. Explain why ...
 2.3.36E: Suppose a linear transformation T : ?n? ?n has the property that T ...
 2.3.37E: Suppose T and U are linear transformations from Is it true that Why...
 2.3.38E: Let T : ?n? ?n be an invertible linear transformation, and let S an...
 2.3.39E: Let T be a linear transformation that maps ?n onto ?n. Show that T ...
 2.3.40E: Suppose T and S satisfy the invertibility equations (1) and (2), wh...
 2.3.41E: [M] Suppose an experiment leads to the following system of equation...
 2.3.42E: Exercises 42–44 show how to use the condition number of a matrix A ...
 2.3.43E: Exercises 42–44 show how to use the condition number of a matrix A ...
 2.3.44E: Exercises 42–44 show how to use the condition number of a matrix A ...
 2.3.45E: [M] Some matrix programs, such as MATLAB, have a command to create ...
Solutions for Chapter 2.3: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 2.3
Get Full SolutionsSince 45 problems in chapter 2.3 have been answered, more than 61156 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 and Its Applications, edition: 4. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178. Chapter 2.3 includes 45 full stepbystep solutions.

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.

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.)

Characteristic equation det(A  AI) = O.
The n roots are the eigenvalues of A.

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.

Column space C (A) =
space of all combinations of the columns of A.

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.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex 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.

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.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

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

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

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

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 ().

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

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.