 4.8.1: For 12, (a) determine a basis for rowspace(A) and make a sketch of ...
 4.8.2: For 12, (a) determine a basis for rowspace(A) and make a sketch of ...
 4.8.3: For 39, (a) find n such that rowspace(A) is a subspace of Rn, and d...
 4.8.4: For 39, (a) find n such that rowspace(A) is a subspace of Rn, and d...
 4.8.5: For 39, (a) find n such that rowspace(A) is a subspace of Rn, and d...
 4.8.6: For 39, (a) find n such that rowspace(A) is a subspace of Rn, and d...
 4.8.7: For 39, (a) find n such that rowspace(A) is a subspace of Rn, and d...
 4.8.8: For 39, (a) find n such that rowspace(A) is a subspace of Rn, and d...
 4.8.9: For 39, (a) find n such that rowspace(A) is a subspace of Rn, and d...
 4.8.10: For 1013, determine a basis for the subspace of Rn spanned by the g...
 4.8.11: For 1013, determine a basis for the subspace of Rn spanned by the g...
 4.8.12: For 1013, determine a basis for the subspace of Rn spanned by the g...
 4.8.13: For 1013, determine a basis for the subspace of Rn spanned by the g...
 4.8.14: Let A = 124 5 11 21 3 7 13 . (a) Find a basis for rowspace(A) and c...
 4.8.15: Give an example of a square matrix A whose row space and column spa...
 4.8.16: Give examples to show how each type of elementary row operation app...
 4.8.17: Let A be an m n matrix with colspace(A) = nullspace(A). Prove that ...
 4.8.18: Let A be an n n matrix with rowspace(A) = nullspace(A). Prove that ...
Solutions for Chapter 4.8: Row Space and Column Space
Full solutions for Differential Equations  4th Edition
ISBN: 9780321964670
Solutions for Chapter 4.8: Row Space and Column Space
Get Full SolutionsThis textbook survival guide was created for the textbook: Differential Equations, edition: 4. Differential Equations was written by and is associated to the ISBN: 9780321964670. Chapter 4.8: Row Space and Column Space includes 18 full stepbystep solutions. Since 18 problems in chapter 4.8: Row Space and Column Space have been answered, more than 20211 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

CayleyHamilton Theorem.
peA) = det(A  AI) has peA) = zero matrix.

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.

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.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

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.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

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.

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

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

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Orthogonal matrix Q.
Square matrix with orthonormal columns, so QT = Ql. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

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 ().

Solvable system Ax = b.
The right side b is in the column space of A.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.

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