 8.1: Let A be an matrix, B a nonzero matrix, and x a vector in expressed...
 8.2: Let (a) Show that (b) Based on your answer to part (a), make a gues...
 8.3: Let be defined by . Show that T is not a linear operator on V.
 8.4: Let be fixed vectors in , and let be the function defined by , wher...
 8.5: Let be the standard basis for , and let be the linear transformatio...
 8.6: Suppose that vectors in are denoted by matrices, and define by (a) ...
 8.7: Let be a basis for a vector space V, and let be the linear operator...
 8.8: Let V and W be vector spaces, let T, , and be linear transformation...
 8.9: Let A and B be similar matrices. Prove: (a) and are similar. (b) If...
 8.10: Fredholm Alternative Theorem Let be a linear operator on an ndimen...
 8.11: Let be the linear operator defined by Find the rank and nullity of T.
 8.12: Prove: If A and B are similar matrices, and if B and C are also sim...
 8.13: Let be the linear operator that is defined by . Find the matrix for...
 8.14: Let and be bases for a vector space V, and let be the transition ma...
 8.15: Let be a basis for a vector space V, and let be a linear operator f...
 8.16: Show that the matrices are similar but that are not.
 8.17: Suppose that is a linear operator, and B is a basis for V for which...
 8.18: Let be a linear operator. Prove that T is onetoone if and only if .
 8.19: (Calculus required) (a) Show that if is twice differentiable, then ...
 8.20: Let be the function defined by the formula (a) Find . (b) Show that...
 8.21: Let , , and be distinct real numbers such that and let be the funct...
 8.22: (Calculus required) Let and be continuous functions, and let V be t...
 8.23: Calculus required Let be the differentiation operator . Show that t...
 8.24: Calculus required It can be shown that for any real number c, the v...
 8.25: Calculus required be the integration transformation defined by wher...
Solutions for Chapter 8: Linear Transformation
Full solutions for Elementary Linear Algebra: Applications Version  10th Edition
ISBN: 9780470432051
Solutions for Chapter 8: Linear Transformation
Get Full SolutionsChapter 8: Linear Transformation includes 25 full stepbystep solutions. Elementary Linear Algebra: Applications Version was written by and is associated to the ISBN: 9780470432051. This textbook survival guide was created for the textbook: Elementary Linear Algebra: Applications Version, edition: 10. Since 25 problems in chapter 8: Linear Transformation have been answered, more than 15436 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions.

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.

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).

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.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

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

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.

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.

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

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.

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.

Outer product uv T
= column times row = rank one matrix.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

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

Saddle point of I(x}, ... ,xn ).
A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

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

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

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).