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Solutions for Chapter 4.4: Orthogonal Bases and Gram-Schmidt

Full solutions for Introduction to Linear Algebra | 4th Edition

ISBN: 9780980232714

Solutions for Chapter 4.4: Orthogonal Bases and Gram-Schmidt

Solutions for Chapter 4.4
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Textbook: Introduction to Linear Algebra
Edition: 4
Author: Gilbert Strang
ISBN: 9780980232714

This textbook survival guide was created for the textbook: Introduction to Linear Algebra, edition: 4. Chapter 4.4: Orthogonal Bases and Gram-Schmidt includes 37 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. Since 37 problems in chapter 4.4: Orthogonal Bases and Gram-Schmidt have been answered, more than 8145 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
  • Adjacency matrix of a graph.

    Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

  • Back substitution.

    Upper triangular systems are solved in reverse order Xn to Xl.

  • Big formula for n by n determinants.

    Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or - sign.

  • Cofactor Cij.

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

  • Complex conjugate

    z = a - ib for any complex number z = a + ib. Then zz = Iz12.

  • Condition number

    cond(A) = c(A) = IIAIlIIA-III = amaxlamin. In Ax = b, the relative change Ilox III Ilx II is less than cond(A) times the relative change Ilob III lib II· Condition numbers measure the sensitivity of the output to change in the input.

  • Diagonal matrix D.

    dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.

  • Diagonalization

    A = S-1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k S-I.

  • 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!).

  • Gram-Schmidt orthogonalization A = QR.

    Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

  • Identity matrix I (or In).

    Diagonal entries = 1, off-diagonal entries = 0.

  • Jordan form 1 = M- 1 AM.

    If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

  • Kirchhoff's Laws.

    Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

  • Network.

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

  • Normal matrix.

    If N NT = NT N, then N has orthonormal (complex) eigenvectors.

  • Outer product uv T

    = column times row = rank one matrix.

  • Simplex method for linear programming.

    The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

  • 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.

  • Special solutions to As = O.

    One free variable is Si = 1, other free variables = o.

  • 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.

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