 3.6.1: . For each of the following matrices, find a basis for the row spac...
 3.6.2: In each of the following, determine the dimension of the subspace o...
 3.6.3: . Let A = 122314 245549 367859 (a) Compute the reduced row echelon ...
 3.6.4: For each of the following choices of A and b, determine whether b i...
 3.6.5: For each consistent system in Exercise 4, determine whether there w...
 3.6.6: How many solutions will the linear system Ax = b have if b is in th...
 3.6.7: Let A be a 6 n matrix of rank r and let b be a vector in R6. For ea...
 3.6.8: Let A be an m n matrix with m > n. Let b Rm and suppose that N(A) =...
 3.6.9: Let A and B be 6 5 matrices. If dim N(A) = 2, what is the rank of A...
 3.6.10: Let A be an m n matrix whose rank is equal to n. If Ac = Ad, does t...
 3.6.11: Let A be an m n matrix. Prove that rank(A) min(m, n)
 3.6.12: Let A and B be row equivalent matrices. (a) Show that the dimension...
 3.6.13: Let A be a 4 3 matrix and suppose that the vectors z1 = 1 1 2 , z2 ...
 3.6.14: Let A be a 4 4 matrix with reduced row echelon form given by U = 10...
 3.6.15: Let A be a 4 5 matrix and let U be the reduced row echelon form of ...
 3.6.16: Let A be a 5 8 matrix with rank equal to 5 and let b be any vector ...
 3.6.17: Let A be a 4 5 matrix. If a1, a2, and a4 are linearly independent a...
 3.6.18: Let A be a 5 3 matrix of rank 3 and let {x1, x2, x3} be a basis for...
 3.6.19: Let A be an mn matrix with rank equal to n. Show that if x = 0 and ...
 3.6.20: Prove that a linear system Ax = b is consistent if and only if the ...
 3.6.21: Let A and B be m n matrices. Show that rank(A + B) rank(A) + rank(B)
 3.6.22: Let A be an m n matrix. (a) Show that if B is a nonsingular m m mat...
 3.6.23: Prove Corollary 3.6.4.
 3.6.24: Show that if A and B are n n matrices and N(A B) = Rn then A = B.
 3.6.25: Let A and B be n n matrices. (a) Show that AB = O if and only if th...
 3.6.26: Let A Rmn and b Rm, and let x0 be a particular solution of the syst...
 3.6.27: Let x and y be nonzero vectors in Rm and Rn, respectively, and let ...
 3.6.28: Let A Rmn, B Rnr , and C = AB. Show that (a) the column space of C ...
 3.6.29: Let A Rmn, B Rnr , and C = AB. Show that (a) if A and B both have l...
 3.6.30: Let A Rmn, B Rnr , and C = AB. Show that (a) if the column vectors ...
 3.6.31: An m n matrix A is said to have a right inverse if there exists an ...
 3.6.32: Prove: If A is an m n matrix and the column vectors of A span Rm, t...
 3.6.33: Show that a matrix B has a left inverse if and only if BT has a rig...
 3.6.34: Let B be an n m matrix whose columns are linearly independent. Show...
 3.6.35: Prove that if a matrix B has a left inverse then the columns of B a...
 3.6.36: Show that if a matrix U is in row echelon form, then the nonzero ro...
Solutions for Chapter 3.6: Row Space and Column Space
Full solutions for Linear Algebra with Applications  9th Edition
ISBN: 9780321962218
Solutions for Chapter 3.6: Row Space and Column Space
Get Full SolutionsLinear Algebra with Applications was written by and is associated to the ISBN: 9780321962218. Chapter 3.6: Row Space and Column Space includes 36 full stepbystep solutions. Since 36 problems in chapter 3.6: Row Space and Column Space have been answered, more than 11888 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 with Applications, edition: 9.

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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.

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

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.

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

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

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.

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

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

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

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

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.