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# Solutions for Chapter 3: Matrices

## Full solutions for Finite Mathematics, Binder Ready Version: An Applied Approach | 11th Edition

ISBN: 9780470876398

Solutions for Chapter 3: Matrices

Solutions for Chapter 3
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##### ISBN: 9780470876398

Finite Mathematics, Binder Ready Version: An Applied Approach was written by and is associated to the ISBN: 9780470876398. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Finite Mathematics, Binder Ready Version: An Applied Approach, edition: 11. Chapter 3: Matrices includes 12 full step-by-step solutions. Since 12 problems in chapter 3: Matrices have been answered, more than 18117 students have viewed full step-by-step solutions from this chapter.

Key Math Terms and definitions covered in this textbook
• Affine transformation

Tv = Av + Vo = linear transformation plus shift.

• Back substitution.

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

• 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.

• Circulant matrix C.

Constant diagonals wrap around as in cyclic shift S. Every circulant is Col + CIS + ... + Cn_lSn - l . Cx = convolution c * x. Eigenvectors in F.

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

Use AT for complex A.

• Gauss-Jordan method.

Invert A by row operations on [A I] to reach [I A-I].

• Minimal polynomial of A.

The lowest degree polynomial with meA) = zero matrix. This is peA) = det(A - AI) if no eigenvalues are repeated; always meA) divides peA).

• Outer product uv T

= column times row = rank one matrix.

• Pascal matrix

Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).

• 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.

• Pivot.

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

• Rotation matrix

R = [~ CS ] rotates the plane by () and R- 1 = RT rotates back by -(). Eigenvalues are eiO and e-iO , eigenvectors are (1, ±i). c, s = cos (), sin ().

• Saddle point of I(x}, ... ,xn ).

A point where the first derivatives of I are zero and the second derivative matrix (a2 II aXi ax j = Hessian matrix) is indefinite.

• 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)!

• Sum V + W of subs paces.

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

• Toeplitz matrix.

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

• 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.

• Tridiagonal matrix T: tij = 0 if Ii - j I > 1.

T- 1 has rank 1 above and below diagonal.

• Unitary matrix UH = U T = U-I.

Orthonormal columns (complex analog of Q).

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