 7.3.1: Label the following statements as true or false. Assume that all ve...
 7.3.2: Label the following statements as true or false. Assume that all ve...
 7.3.3: For each linear operator T on V, find the minimal polynomial of T. ...
 7.3.4: Determine which of the matrices and operators in Exercises 2 and 3 ...
 7.3.5: Describe all linear operators T on R2 such that T is diagonalizable...
 7.3.6: Prove Theorem 7.13 and its corollary.
 7.3.7: Prove the corollary to Theorem 7.14.
 7.3.8: Let T be a linear operator on a finitedimensional vector space, an...
 7.3.9: Let T be a diagonalizable linear operator on a finitedimensional v...
 7.3.10: Let T be a linear operator on a finitedimensional vector space V, ...
 7.3.11: Let g(t) be the auxiliary polynomial associated with a homogeneous ...
 7.3.12: Let D be the differentiation operator on P(/?), the space of polyno...
 7.3.13: Let T be a linear operator on a finitedimensional vector space, an...
 7.3.14: Let T be linear operator on a finitedimensional vector space V, an...
 7.3.15: Let T be a linear operator on a finitedimensional vector space V, ...
 7.3.16: T be a linear operator on a finitedimensional vector space V, and ...
Solutions for Chapter 7.3: The Minimal Polynomial
Full solutions for Linear Algebra  4th Edition
ISBN: 9780130084514
Solutions for Chapter 7.3: The Minimal Polynomial
Get Full SolutionsChapter 7.3: The Minimal Polynomial includes 16 full stepbystep solutions. Linear Algebra was written by and is associated to the ISBN: 9780130084514. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Linear Algebra , edition: 4. Since 16 problems in chapter 7.3: The Minimal Polynomial have been answered, more than 12069 students have viewed full stepbystep solutions from this chapter.

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

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.)

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

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).

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.

Dimension of vector space
dim(V) = number of vectors in any basis for V.

Hypercube matrix pl.
Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

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.

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

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

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

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

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

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.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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.