 2.1.1: Exer. 144: Solve the equation.
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 2.1.45: Exer. 4550: Show that the equation is an identity.
 2.1.46: Exer. 4550: Show that the equation is an identity.
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 2.1.51: Exer. 5152: For what value of c is the number a a solution of the e...
 2.1.52: Exer. 5152: For what value of c is the number a a solution of the e...
 2.1.53: Exer. 5354: Determine whether the two equations are equivalent.
 2.1.54: Exer. 5354: Determine whether the two equations are equivalent.
 2.1.55: Exer. 5556: Determine values for a and b such that is a solution of...
 2.1.56: Exer. 5556: Determine values for a and b such that is a solution of...
 2.1.57: Exer. 5758: Determine which equation is not equivalent to the equat...
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 2.1.59: Exer. 5962: Solve the formula for the specified variable.
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 2.1.61: Exer. 5962: Solve the formula for the specified variable.
 2.1.62: Exer. 5962: Solve the formula for the specified variable.
 2.1.63: Exer. 6376: The formula occurs in the indicated application. Solve ...
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Solutions for Chapter 2.1: Equations
Full solutions for Algebra and Trigonometry with Analytic Geometry  12th Edition
ISBN: 9780495559719
Solutions for Chapter 2.1: Equations
Get Full SolutionsChapter 2.1: Equations includes 76 full stepbystep solutions. Algebra and Trigonometry with Analytic Geometry was written by and is associated to the ISBN: 9780495559719. Since 76 problems in chapter 2.1: Equations have been answered, more than 37630 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Algebra and Trigonometry with Analytic Geometry, edition: 12. This expansive textbook survival guide covers the following chapters and their solutions.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

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.

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.)

Complete solution x = x p + Xn to Ax = b.
(Particular x p) + (x n in nullspace).

Elimination.
A sequence of row operations that reduces A to an upper triangular U or to the reduced form R = rref(A). Then A = LU with multipliers eO in L, or P A = L U with row exchanges in P, or E A = R with an invertible E.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

Full column rank r = n.
Independent columns, N(A) = {O}, no free variables.

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

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.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

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).

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Schur complement S, D  C A } B.
Appears in block elimination on [~ g ].

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

Vector addition.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.