 4.2.1E: Determine if is in Nul A, where
 4.2.2E: Determine if w = is in Nul A, where A =
 4.2.3E: Find an explicit description of Nul A by listing vectors that span ...
 4.2.4E: Find an explicit description of Nul A by listing vectors that span ...
 4.2.5E: Find an explicit description of Nul A by listing vectors that span ...
 4.2.6E: Find an explicit description of Nul A by listing vectors that span ...
 4.2.7E: In Exercises 7–14, either use an appropriate theorem to show that t...
 4.2.8E: Either use an appropriate theorem to show that the given set, W, is...
 4.2.9E: Either use an appropriate theorem to show that the given set, W, is...
 4.2.10E: Either use an appropriate theorem to show that the given set, W, is...
 4.2.11E: Either use an appropriate theorem to show that the given set, W, is...
 4.2.12E: Either use an appropriate theorem to show that the given set, W, is...
 4.2.13E: In Exercises 7–14, either use an appropriate theorem to show that t...
 4.2.14E: Either use an appropriate theorem to show that the given set, W, is...
 4.2.15E: Find A such that the given set is Col A.
 4.2.16E: Find A such that the given set is Col A.
 4.2.17E: (a) find k such that Nul A is a subspace of ?k, and (b) find k such...
 4.2.18E: (a) find k such that Nul A is a subspace of ?k, and (b) find k such...
 4.2.19E: For the matrices in Exercises 17–20, (a) find ksuch that Nul A is a...
 4.2.20E: (a) find k such that Nul A is a subspace of ?k, and (b) find k such...
 4.2.21E: With A as in Exercise 17, find a nonzero vector in Nul A and a nonz...
 4.2.22E: With A as in Exercise 3, find a nonzero vector in Nul A and a nonze...
 4.2.23E: Let and Determine if w is in Col A. Is w in Nul A?
 4.2.24E: Let and Determine if w is in Col A. Is w in Nul A
 4.2.25E: In Exercises 25 and 26, A denotes an m × n matrix. Mark each statem...
 4.2.26E: In Exercises 25 and 26, A denotes an m × n matrix. Mark each statem...
 4.2.27E: It can be shown that a solution of the system below is Use this fac...
 4.2.28E: Consider the following two systems of equations: It can be shown th...
 4.2.29E: Prove Theorem 3 as follows: Given an m × n matrix A, an element in ...
 4.2.30E: be a linear transformation from a vector space V into a vector spac...
 4.2.31E: Define For instance, if a. Show that T is a linear transformation. ...
 4.2.32E: Define a linear transformation Find polynomials p1 and p2 in that s...
 4.2.33E: Let M2×2 be the vector space of all 2 × 2 matrices, and define a. S...
 4.2.34E: (Calculus required) Define as follows:For be the antiderivative F o...
 4.2.35E: Let V and W be vector spaces, and let be a linear transformation. G...
 4.2.36E: Given as in Exercise 35, and given a subspace Z of W, let U be the ...
 4.2.37E: [M] Determine whether w is in the column space of A, the null space...
 4.2.38E: [M] Determine whether w is in the column space of A, the null space...
 4.2.39E: denote the columns of the matrix A, where a. Explain why a3 and a5 ...
 4.2.40E: Then H and K are subspaces of . In fact, H and K are planes in thro...
Solutions for Chapter 4.2: Linear Algebra and Its Applications 5th Edition
Full solutions for Linear Algebra and Its Applications  5th Edition
ISBN: 9780321982384
Solutions for Chapter 4.2
Get Full SolutionsLinear Algebra and Its Applications was written by and is associated to the ISBN: 9780321982384. Since 40 problems in chapter 4.2 have been answered, more than 41243 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: 5. Chapter 4.2 includes 40 full stepbystep solutions.

Affine transformation
Tv = Av + Vo = linear transformation plus shift.

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.

Cholesky factorization
A = CTC = (L.J]))(L.J]))T for positive definite A.

Companion matrix.
Put CI, ... ,Cn in row n and put n  1 ones just above the main diagonal. Then det(A  AI) = ±(CI + c2A + C3A 2 + .•. + cnA nl  An).

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.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

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.

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

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.

Iterative method.
A sequence of steps intended to approach the desired solution.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

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

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

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).

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.

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

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·