 1.4.1E: Compute the products in Exercises 1–4 using (a) the definition, as ...
 1.4.2E: Compute the products in Exercises 1–4 using (a) the definition, as ...
 1.4.3E: Compute the products in Exercises 1–4 using (a) the definition, as ...
 1.4.4E: Compute the products in Exercises 1–4 using (a) the definition, as ...
 1.4.5E: In Exercises 5–8, use the definition of Ax to write the matrix equa...
 1.4.6E: In Exercises 5–8, use the definition of Ax to write the matrix equa...
 1.4.7E: In Exercises 5–8, use the definition of Ax to write the matrix equa...
 1.4.8E: In Exercises 5–8, use the definition of Ax to write the matrix equa...
 1.4.9E: In Exercises 9 and 10, write the system first as a vector equation ...
 1.4.10E: In Exercises 9 and 10, write the system first as a vector equation ...
 1.4.11E: Given A and b in Exercises 11 and 12, write the augmented matrix fo...
 1.4.12E: Given A and b in Exercises 11 and 12, write the augmented matrix fo...
 1.4.13E: Is u in the plane in ?3 spanned by the columns of A? (See the figur...
 1.4.14E: Is u in the subset of ?3 spanned by the columns of A? Why or why not?
 1.4.15E: Show that the equation Ax = b does not have a solution for all poss...
 1.4.16E: Repeat the requests from Exercise 15 with Exercise 15: Show that th...
 1.4.17E: Exercises 17–20 refer to the matrices A and B below. Make appropria...
 1.4.18E: Exercises 17–20 refer to the matrices A and B below. Make appropria...
 1.4.19E: Exercises 17–20 refer to the matrices A and B below. Make appropria...
 1.4.20E: Exercises 17–20 refer to the matrices A and B below. Make appropria...
 1.4.21E: Does {v1, v2, v3} span ?4 ? Why or why not?
 1.4.22E: Does {v1, v2, v3} span ?3 ? Why or why not?
 1.4.23E: a. The equation Ax = b is referred to as a vector equation.b. A vec...
 1.4.24E: a. Every matrix equation Ax = b corresponds to a vector equation wi...
 1.4.25E: Note that Use this fact (and no row operations) to find scalars c1,...
 1.4.26E: It can be shown that 2u – 3v – w = 0. Use this fact (and no row ope...
 1.4.27E: Rewrite the (numerical) matrix equation below in symbolic form as a...
 1.4.28E: Let q1, q2, q3, and v represent vectors in ?5, and let x1, x2, and ...
 1.4.29E: Construct a 3 × 3 matrix, not in echelon form, whose columns span ?...
 1.4.30E: Construct a 3 × 3 matrix, not in echelon form, whose columns do not...
 1.4.31E: Let A be a 3 × 2 matrix. Explain why the equation Ax = b cannot be ...
 1.4.32E: Could a set of three vectors in R4 span all of ?4? Explain. What ab...
 1.4.33E: Suppose A is a 4 × 3 matrix and b is a vector in ?4 with the proper...
 1.4.34E: Let A be a 3 × 4 matrix, let v1 and v2 be vectors in ?3, and let w ...
 1.4.35E: Let A be a 5 × 3 matrix, let y be a vector in ?3, and let z be a ve...
 1.4.36E: Suppose A is a 4 × 4 matrix and b is a vector in ?4, with the prope...
 1.4.37E: [M] In Exercises 37–40, determine if the columns of the matrix span...
 1.4.38E: [M] In Exercises 37–40, determine if the columns of the matrix span...
 1.4.39E: [M] In Exercises 37–40, determine if the columns of the matrix span...
 1.4.40E: [M] In Exercises 37–40, determine if the columns of the matrix span...
 1.4.41E: [M] Find a column of the matrix in Exercise 39 that can be deleted ...
 1.4.42E: [M] Find a column of the matrix in Exercise 40 that can be deleted ...
Solutions for Chapter 1.4: Linear Algebra and Its Applications 4th Edition
Full solutions for Linear Algebra and Its Applications  4th Edition
ISBN: 9780321385178
Solutions for Chapter 1.4
Get Full SolutionsSince 42 problems in chapter 1.4 have been answered, more than 30994 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.4 includes 42 full stepbystep 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.

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.

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

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

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

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

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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

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

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).

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

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

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

Pascal matrix
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

Pivot columns of A.
Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

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

Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b  Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) 1 AT.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).