×
Log in to StudySoup
Get Full Access to Math - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Math - Textbook Survival Guide

Solutions for Chapter 4.7: Coordinates and Change of Basis

Elementary Linear Algebra | 8th Edition | ISBN: 9781305658004 | Authors: Ron Larson

Full solutions for Elementary Linear Algebra | 8th Edition

ISBN: 9781305658004

Elementary Linear Algebra | 8th Edition | ISBN: 9781305658004 | Authors: Ron Larson

Solutions for Chapter 4.7: Coordinates and Change of Basis

Solutions for Chapter 4.7
4 5 0 326 Reviews
22
3
Textbook: Elementary Linear Algebra
Edition: 8
Author: Ron Larson
ISBN: 9781305658004

Chapter 4.7: Coordinates and Change of Basis includes 116 full step-by-step solutions. Since 116 problems in chapter 4.7: Coordinates and Change of Basis have been answered, more than 33749 students have viewed full step-by-step solutions from this chapter. Elementary Linear Algebra was written by and is associated to the ISBN: 9781305658004. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Elementary Linear Algebra, edition: 8.

Key Math Terms and definitions covered in this textbook
  • Augmented matrix [A b].

    Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

  • 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.

  • Cayley-Hamilton Theorem.

    peA) = det(A - AI) has peA) = zero matrix.

  • Cross product u xv in R3:

    Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

  • Elimination matrix = Elementary matrix Eij.

    The identity matrix with an extra -eij in the i, j entry (i #- j). Then Eij A subtracts eij times row j of A from row i.

  • Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).

    Use AT for complex A.

  • Full column rank r = n.

    Independent columns, N(A) = {O}, no free variables.

  • Hypercube matrix pl.

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

  • Iterative method.

    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.

  • Linearly dependent VI, ... , Vn.

    A combination other than all Ci = 0 gives L Ci Vi = O.

  • Network.

    A directed graph that has constants Cl, ... , Cm associated with the edges.

  • Orthonormal vectors q 1 , ... , q n·

    Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

  • Projection p = a(aTblaTa) onto the line through a.

    P = aaT laTa has rank l.

  • Pseudoinverse A+ (Moore-Penrose inverse).

    The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

  • Spanning set.

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

  • Subspace S of V.

    Any vector space inside V, including V and Z = {zero vector only}.

  • Sum V + W of subs paces.

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

  • Transpose matrix AT.

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

  • Volume of box.

    The rows (or the columns) of A generate a box with volume I det(A) I.

×
Log in to StudySoup
Get Full Access to Math - Textbook Survival Guide
Join StudySoup for FREE
Get Full Access to Math - Textbook Survival Guide
×
Reset your password