- 5.2.1: The system of equations describes the distance of two hikers, Edna ...
- 5.2.2: Check that each ordered pair is a solution to each system. If the p...
- 5.2.3: Solve each equation by symbolic manipulation. a.14 + 2x = 4 3x b. 7...
- 5.2.4: Solve the system of equations using the substitution method, and ch...
- 5.2.5: Substitute 4 3x for y. Then rewrite each expression in terms of one...
- 5.2.6: Solve each system of equations by substitution, and check your solu...
- 5.2.7: APPLICATION This system of equations models the profits of two home...
- 5.2.8: The costs for two families to attend Friday nights basketball game ...
- 5.2.9: APPLICATION The manager of a movie theater wants to know the number...
- 5.2.10: Students in an algebra class did an experiment similar to the Inves...
- 5.2.11: The table at right gives the equations that model the three vehicle...
- 5.2.12: APPLICATION This table shows the winning times for the Olympic wome...
- 5.2.13: A candy store manager is making a sour candy mix by combining sour ...
- 5.2.14: Mrs. Abdul mixes bottled fruit juice with natural orange soda to ma...
- 5.2.15: A system of two linear equations has the solution (3, 4.5).Write th...
- 5.2.16: You and your family are visiting Seattle and take the elevator to t...
- 5.2.17: Do each calculation by hand, and then check your results with a cal...
- 5.2.18: Match each matrix multiplication with its answer.
Solutions for Chapter 5.2: Solving Systems of Equations Using Substitution
Full solutions for Discovering Algebra: An Investigative Approach | 2nd Edition
Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.
Upper triangular systems are solved in reverse order Xn to Xl.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Determinant IAI = det(A).
Defined by det I = 1, sign reversal for row exchange, and linearity in each row. Then IAI = 0 when A is singular. Also IABI = IAIIBI and
Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
Dimension of vector space
dim(V) = number of vectors in any basis for V.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA-1 yll2 = Y T(AAT)-1 Y = 1 displayed by eigshow; axis lengths ad
Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n - 1)/2 edges between nodes. A tree has only n - 1 edges and no closed loops.
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.
lA-II = 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.
Markov matrix M.
All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.
Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.
Pseudoinverse A+ (Moore-Penrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).
Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.
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.
Solvable system Ax = b.
The right side b is in the column space of A.
Vandermonde matrix V.
V c = b gives coefficients of p(x) = Co + ... + Cn_IXn- 1 with P(Xi) = bi. Vij = (Xi)j-I and det V = product of (Xk - Xi) for k > i.
Volume of box.
The rows (or the columns) of A generate a box with volume I det(A) I.
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