- 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 n-dimen...
- 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 one-to-one 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
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.
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).
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 IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Identity matrix I (or In).
Diagonal entries = 1, off-diagonal 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 A-I.
Square matrix with A-I A = I and AA-l = 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 B-1 A-I and (A-I)T. Cofactor formula (A-l)ij = Cji! detA.
A sequence of steps intended to approach the desired solution.
Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , Aj-Ib. 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.
lA-II = 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 = U-I.
Orthonormal columns (complex analog of Q).