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# Solutions for Chapter 1.10: Linear Algebra and Its Applications 4th Edition

## Full solutions for Linear Algebra and Its Applications | 4th Edition

ISBN: 9780321385178

Solutions for Chapter 1.10

Solutions for Chapter 1.10
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##### ISBN: 9780321385178

This textbook survival guide was created for the textbook: Linear Algebra and Its Applications, edition: 4. Linear Algebra and Its Applications was written by and is associated to the ISBN: 9780321385178. This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.10 includes 14 full step-by-step solutions. Since 14 problems in chapter 1.10 have been answered, more than 30260 students have viewed full step-by-step solutions from this chapter.

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.)

• Column space C (A) =

space of all combinations of the columns of A.

• Cramer's Rule for Ax = b.

B j has b replacing column j of A; x j = det B j I det A

• Diagonalizable matrix A.

Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then S-I AS = A = eigenvalue matrix.

• 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.

• Kronecker product (tensor product) A ® B.

Blocks aij B, eigenvalues Ap(A)Aq(B).

• Length II x II.

Square root of x T x (Pythagoras in n dimensions).

• Multiplier eij.

The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

• Nullspace matrix N.

The columns of N are the n - r special solutions to As = O.

• Particular solution x p.

Any solution to Ax = b; often x p has free variables = o.

• 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.

• Reflection matrix (Householder) Q = I -2uuT.

Unit vector u is reflected to Qu = -u. All x intheplanemirroruTx = o have Qx = x. Notice QT = Q-1 = Q.

• Row space C (AT) = all combinations of rows of A.

Column vectors by convention.

• Schwarz inequality

Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

• Singular matrix A.

A square matrix that has no inverse: det(A) = o.

• Spanning set.

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

• Spectral Theorem A = QAQT.

Real symmetric A has real A'S and orthonormal q's.

• Subspace S of V.

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