 8.1  8.3.8.18.38.4: In Exercises 16, determine whether each relation is a function. Giv...
 8.1  8.3.8.18.38.5: In Exercises 16, determine whether each relation is a function. Giv...
 8.1  8.3.8.18.38.6: In Exercises 16, determine whether each relation is a function. Giv...
 8.1  8.3.8.18.38.7: In Exercises 16, determine whether each relation is a function. Giv...
 8.1  8.3.8.18.38.8: In Exercises 16, determine whether each relation is a function. Giv...
 8.1  8.3.8.18.38.9: In Exercises 16, determine whether each relation is a function. Giv...
 8.1  8.3.8.18.38.10: Use the graph of f to solve Exercises 712. Explain why f represents...
 8.1  8.3.8.18.38.11: Use the graph of f to solve Exercises 712. Use the graph to find f(4).
 8.1  8.3.8.18.38.12: Use the graph of f to solve Exercises 712. For what value or values...
 8.1  8.3.8.18.38.13: Use the graph of f to solve Exercises 712. For what value or values...
 8.1  8.3.8.18.38.14: Use the graph of f to solve Exercises 712. Find the domain of f.
 8.1  8.3.8.18.38.15: Use the graph of f to solve Exercises 712. Find the range of f.
 8.1  8.3.8.18.38.16: In Exercises 1314, find the domain of each function f(x) (x 2)(x 2)
 8.1  8.3.8.18.38.17: In Exercises 1314, find the domain of each function g(x) 1 (x 2)(x ...
 8.1  8.3.8.18.38.18: In Exercises 1522, let f(x) x2 3x 8 and g(x) 2x 5. Find each of the...
 8.1  8.3.8.18.38.19: In Exercises 1522, let f(x) x2 3x 8 and g(x) 2x 5. Find each of the...
 8.1  8.3.8.18.38.20: In Exercises 1522, let f(x) x2 3x 8 and g(x) 2x 5. Find each of the...
 8.1  8.3.8.18.38.21: In Exercises 1522, let f(x) x2 3x 8 and g(x) 2x 5. Find each of the...
 8.1  8.3.8.18.38.22: In Exercises 1522, let f(x) x2 3x 8 and g(x) 2x 5. Find each of the...
 8.1  8.3.8.18.38.23: In Exercises 1522, let f(x) x2 3x 8 and g(x) 2x 5. Find each of the...
 8.1  8.3.8.18.38.24: In Exercises 1522, let f(x) x2 3x 8 and g(x) 2x 5. Find each of the...
 8.1  8.3.8.18.38.25: In Exercises 1522, let f(x) x2 3x 8 and g(x) 2x 5. Find each of the...
Solutions for Chapter 8.1  8.3: MIDCHAPTER CHECK POINT
Full solutions for Introductory & Intermediate Algebra for College Students  4th Edition
ISBN: 9780321758941
Solutions for Chapter 8.1  8.3: MIDCHAPTER CHECK POINT
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Introductory & Intermediate Algebra for College Students was written by and is associated to the ISBN: 9780321758941. Chapter 8.1  8.3: MIDCHAPTER CHECK POINT includes 22 full stepbystep solutions. Since 22 problems in chapter 8.1  8.3: MIDCHAPTER CHECK POINT have been answered, more than 68690 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Introductory & Intermediate Algebra for College Students, edition: 4.

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.

Condition number
cond(A) = c(A) = IIAIlIIAIII = 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.

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.

Full row rank r = m.
Independent rows, at least one solution to Ax = b, column space is all of Rm. Full rank means full column rank or full row rank.

Indefinite matrix.
A symmetric matrix with eigenvalues of both signs (+ and  ).

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.

Krylov subspace Kj(A, b).
The subspace spanned by b, Ab, ... , AjIb. Numerical methods approximate A I b by x j with residual b  Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

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

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 .

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

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.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Reduced row echelon form R = rref(A).
Pivots = 1; zeros above and below pivots; the r nonzero rows of R give a basis for the row space of A.

Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)l has AA+ = 1m.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

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}.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.