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# Solutions for Chapter 4.9: Rank of a Matrix

## Full solutions for Elementary Linear Algebra with Applications | 9th Edition

ISBN: 9780132296540

Solutions for Chapter 4.9: Rank of a Matrix

Solutions for Chapter 4.9
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##### ISBN: 9780132296540

Chapter 4.9: Rank of a Matrix includes 51 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. Since 51 problems in chapter 4.9: Rank of a Matrix have been answered, more than 58450 students have viewed full step-by-step solutions from this chapter. Elementary Linear Algebra with Applications was written by and is associated to the ISBN: 9780132296540. This textbook survival guide was created for the textbook: Elementary Linear Algebra with Applications, edition: 9.

Key Math Terms and definitions covered in this textbook
• 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.

• 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.)

• Diagonal matrix D.

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

• Echelon matrix U.

The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

• 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.

• Factorization

A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

• 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.

• 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.

• Matrix multiplication AB.

The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

• Orthogonal matrix Q.

Square matrix with orthonormal columns, so QT = Q-l. Preserves length and angles, IIQxll = IIxll and (QX)T(Qy) = xTy. AlllAI = 1, with orthogonal eigenvectors. Examples: Rotation, reflection, permutation.

• Pivot columns of A.

Columns that contain pivots after row reduction. These are not combinations of earlier columns. The pivot columns are a basis for the column space.

• Projection matrix P onto subspace S.

Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.

• Rank r (A)

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

• 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.

• Spanning set.

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

• Spectrum of A = the set of eigenvalues {A I, ... , An}.

Spectral radius = max of IAi I.

• Symmetric factorizations A = LDLT and A = QAQT.

Signs in A = signs in D.

• Toeplitz matrix.

Constant down each diagonal = time-invariant (shift-invariant) filter.

• Vector v in Rn.

Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.