- 5.1.1: Verify whether the given ordered pair is a solution to the system. ...
- 5.1.2: Match each graph of a system of equations with its corresponding ta...
- 5.1.3: Graph each system on your calculator using the window given. Use th...
- 5.1.4: Use the calculator table function to find the solution to each syst...
- 5.1.5: Solve the equations for y, then find the value of y when x = 1. Sub...
- 5.1.6: APPLICATION Two friends start rival Internet companies in their hom...
- 5.1.7: APPLICATION After seeing her friends profit from their websites in ...
- 5.1.8: APPLICATION The total tuition for students at University College an...
- 5.1.9: The high school band and drill team both practice on the football f...
- 5.1.10: The equations y = 28.65 0.0411(x 1962) and y = 27.5 0.0411(x 1990) ...
- 5.1.11: Mini-Investigation Consider the system of equations Explain what va...
- 5.1.12: APPLICATION Hydroplanes are boats that move so fast they skim the t...
- 5.1.13: Solve each equation using the method you like best. Then substitute...
- 5.1.14: Write the equation represented by this balance. Then solve the equa...
- 5.1.15: Find each matrix sum and difference.
- 5.1.16: . Solve each equation for y. a. y + 2 = 5x b. 5y = 4 7x c. 2y 4x = 3
Solutions for Chapter 5.1: Solving Systems of Equations
Full solutions for Discovering Algebra: An Investigative Approach | 2nd Edition
Tv = Av + Vo = linear transformation plus shift.
Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.
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.
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.
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
Free variable Xi.
Column i has no pivot in elimination. We can give the n - r free variables any values, then Ax = b determines the r pivot variables (if solvable!).
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
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.
Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.
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.
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).
A directed graph that has constants Cl, ... , Cm associated with the edges.
Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.
Nullspace N (A)
= All solutions to Ax = O. Dimension n - r = (# columns) - rank.
Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q -1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •
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.
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 matrix A.
A square matrix that has no inverse: det(A) = o.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!
Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.