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Solutions for Chapter 7.2: The Matrix of a Linear Transformation

Full solutions for Introduction to Linear Algebra | 4th Edition

ISBN: 9780980232714

Solutions for Chapter 7.2: The Matrix of a Linear Transformation

Solutions for Chapter 7.2
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Textbook: Introduction to Linear Algebra
Edition: 4
Author: Gilbert Strang
ISBN: 9780980232714

Since 38 problems in chapter 7.2: The Matrix of a Linear Transformation have been answered, more than 8145 students have viewed full step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Introduction to Linear Algebra, edition: 4. Chapter 7.2: The Matrix of a Linear Transformation includes 38 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Introduction to Linear Algebra was written by and is associated to the ISBN: 9780980232714.

Key Math Terms and definitions covered in this textbook
  • 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].

  • Free variable Xi.

    Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

  • Hermitian matrix A H = AT = A.

    Complex analog a j i = aU of a symmetric matrix.

  • Hypercube matrix pl.

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

  • Incidence matrix of a directed graph.

    The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

  • 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.

  • Least squares solution X.

    The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b - Ax is orthogonal to all columns of A.

  • Left nullspace N (AT).

    Nullspace of AT = "left nullspace" of A because y T A = OT.

  • Lucas numbers

    Ln = 2,J, 3, 4, ... satisfy Ln = L n- l +Ln- 2 = A1 +A~, with AI, A2 = (1 ± -/5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

  • Multiplicities AM and G M.

    The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

  • Pivot.

    The diagonal entry (first nonzero) at the time when a row is used in elimination.

  • Plane (or hyperplane) in Rn.

    Vectors x with aT x = O. Plane is perpendicular to a =1= O.

  • Rank one matrix A = uvT f=. O.

    Column and row spaces = lines cu and cv.

  • Rank r (A)

    = number of pivots = dimension of column space = dimension of row space.

  • Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.

    Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

  • 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.

  • Solvable system Ax = b.

    The right side b is in the column space of 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}.

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