 4.7.1: List the row vectors and column vectors of the matrix
 4.7.2: Express the product as a linear combination of the column vectors o...
 4.7.3: Determine whether is in the column space of A, and if so, express a...
 4.7.4: Suppose that , , , is a solution of a nonhomogeneous linear system ...
 4.7.5: In parts (a)(d), find the vector form of the general solution of th...
 4.7.6: Find a basis for the null space of A. (a) (b) (c) (d) (e)
 4.7.7: In each part, a matrix in row echelon form is given. By inspection,...
 4.7.8: For the matrices in Exercise 6, find a basis for the row space of A...
 4.7.9: By inspection, find a basis for the row space and a basis for the c...
 4.7.10: For the matrices in Exercise 6, find a basis for the row space of A...
 4.7.11: Find a basis for the subspace of spanned by the given vectors.
 4.7.12: Find a subset of the vectors that forms a basis for the space spann...
 4.7.13: Prove that the row vectors of an invertible matrix A form a basis for
 4.7.14: Construct a matrix whose null space consists of all linear combinat...
 4.7.15: (a) Let Show that relative to an coordinate system in 3space the ...
 4.7.16: Find a matrix whose null space is (a) a point. (b) a line. (c) a plane
 4.7.17: (a) Find all matrices whose null space is the line (b) Sketch the n...
 4.7.18: The equation can be viewed as a linear system of one equation in th...
 4.7.19: Suppose that A and B are matrices and A is invertible.Invent and pr...
 4.7.a: In parts (a)(j) determine whether the statement is true or false, a...
 4.7.b: In parts (a)(j) determine whether the statement is true or false, a...
 4.7.c: In parts (a)(j) determine whether the statement is true or false, a...
 4.7.d: In parts (a)(j) determine whether the statement is true or false, a...
 4.7.e: In parts (a)(j) determine whether the statement is true or false, a...
 4.7.f: In parts (a)(j) determine whether the statement is true or false, a...
 4.7.g: In parts (a)(j) determine whether the statement is true or false, a...
 4.7.h: In parts (a)(j) determine whether the statement is true or false, a...
 4.7.i: In parts (a)(j) determine whether the statement is true or false, a...
 4.7.j: In parts (a)(j) determine whether the statement is true or false, a...
Solutions for Chapter 4.7: Row Space, Column Space, and Null Space
Full solutions for Elementary Linear Algebra: Applications Version  10th Edition
ISBN: 9780470432051
Solutions for Chapter 4.7: Row Space, Column Space, and Null Space
Get Full SolutionsChapter 4.7: Row Space, Column Space, and Null Space includes 29 full stepbystep solutions. Since 29 problems in chapter 4.7: Row Space, Column Space, and Null Space have been answered, more than 15465 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: Elementary Linear Algebra: Applications Version, edition: 10. Elementary Linear Algebra: Applications Version was written by and is associated to the ISBN: 9780470432051.

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.)

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).

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.

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

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

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

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

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.

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

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

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!

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

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

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

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

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.