 1.1.1: In each exercise set, problems marked with C are designed tobe solv...
 1.1.2: In each exercise set, problems marked with C are designed tobe solv...
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 1.1.6: In each exercise set, problems marked with C are designed tobe solv...
 1.1.7: In Exercises 78, determine which of (a)(d) form a solution tothe gi...
 1.1.8: In Exercises 78, determine which of (a)(d) form a solution tothe gi...
 1.1.9: In Exercises 914, find all solutions to the given system by elimina...
 1.1.10: In Exercises 914, find all solutions to the given system by elimina...
 1.1.11: In Exercises 914, find all solutions to the given system by elimina...
 1.1.12: In Exercises 914, find all solutions to the given system by elimina...
 1.1.13: In Exercises 914, find all solutions to the given system by elimina...
 1.1.14: In Exercises 914, find all solutions to the given system by elimina...
 1.1.15: In Exercises 1522, determine if the given linear system is in echel...
 1.1.16: In Exercises 1522, determine if the given linear system is in echel...
 1.1.17: In Exercises 1522, determine if the given linear system is in echel...
 1.1.18: In Exercises 1522, determine if the given linear system is in echel...
 1.1.19: In Exercises 1522, determine if the given linear system is in echel...
 1.1.20: In Exercises 1522, determine if the given linear system is in echel...
 1.1.21: In Exercises 1522, determine if the given linear system is in echel...
 1.1.22: In Exercises 1522, determine if the given linear system is in echel...
 1.1.23: In Exercises 2330, find the set of solutions for the given linearsy...
 1.1.24: In Exercises 2330, find the set of solutions for the given linearsy...
 1.1.25: In Exercises 2330, find the set of solutions for the given linearsy...
 1.1.26: In Exercises 2330, find the set of solutions for the given linearsy...
 1.1.27: In Exercises 2330, find the set of solutions for the given linearsy...
 1.1.28: In Exercises 2330, find the set of solutions for the given linearsy...
 1.1.29: In Exercises 2330, find the set of solutions for the given linearsy...
 1.1.30: In Exercises 2330, find the set of solutions for the given linearsy...
 1.1.31: In Exercises 3134, each linear system is not in echelon form butcan...
 1.1.32: In Exercises 3134, each linear system is not in echelon form butcan...
 1.1.33: In Exercises 3134, each linear system is not in echelon form butcan...
 1.1.34: In Exercises 3134, each linear system is not in echelon form butcan...
 1.1.35: For what value(s) of k is the linear system consistent?6x1 5x2 = 49...
 1.1.36: For what value(s) of h is the linear system consistent?6x1 8x2 = h9...
 1.1.37: Find values of h andk so that the linear system has no solutions.2x...
 1.1.38: For what values of h and k does the linear system have infinitelyma...
 1.1.39: A system of linear equations is in echelon form. If there arefour f...
 1.1.40: Suppose that a system of five equations with eight unknownsis in ec...
 1.1.41: Suppose that a system of seven equations with thirteen unknownsis i...
 1.1.42: A linear system is in echelon form. There are a total of ninevariab...
 1.1.43: FIND AN EXAMPLE For Exercises 4350, find an example thatmeets the g...
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 1.1.51: TRUE OR FALSE For Exercises 5160, determine if the statementis true...
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 1.1.59: TRUE OR FALSE For Exercises 5160, determine if the statementis true...
 1.1.60: TRUE OR FALSE For Exercises 5160, determine if the statementis true...
 1.1.61: Referring to Example 1, suppose that the minimum outsidetemperature...
 1.1.62: Referring to Example 1, suppose that the minimum outsidetemperature...
 1.1.63: A total of 385 people attend the premiere of a new movie.Ticket pri...
 1.1.64: For tax and accounting purposes, corporations depreciate thevalue o...
 1.1.65: (Calculus required) Suppose that f (x) = a1e2x + a2e3x is asolution...
 1.1.66: An investor has $100,000 and can invest in any combinationof two ty...
 1.1.67: Degrees Fahrenheit (F) and Celsius (C) are related by a linearequat...
 1.1.68: A 60gallon bathtub is to be filled with water that is exactly100F....
 1.1.69: This problem requires about 8 nickels, 8 quarters, and a sheetof 8....
 1.1.70: The Bixby Creek Bridge is located along Californias Big Surcoast an...
 1.1.71: C In Exercises 7176, use computational assistance to find theset of...
 1.1.72: C In Exercises 7176, use computational assistance to find theset of...
 1.1.73: C In Exercises 7176, use computational assistance to find theset of...
 1.1.74: C In Exercises 7176, use computational assistance to find theset of...
 1.1.75: C In Exercises 7176, use computational assistance to find theset of...
 1.1.76: C In Exercises 7176, use computational assistance to find theset of...
Solutions for Chapter 1.1: Lines and Linear Equations
Full solutions for Linear Algebra with Applications  1st Edition
ISBN: 9780716786672
Solutions for Chapter 1.1: Lines and Linear Equations
Get Full SolutionsThis textbook survival guide was created for the textbook: Linear Algebra with Applications, edition: 1. This expansive textbook survival guide covers the following chapters and their solutions. Since 76 problems in chapter 1.1: Lines and Linear Equations have been answered, more than 14582 students have viewed full stepbystep solutions from this chapter. Chapter 1.1: Lines and Linear Equations includes 76 full stepbystep solutions. Linear Algebra with Applications was written by and is associated to the ISBN: 9780716786672.

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.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Diagonalization
A = S1 AS. A = eigenvalue matrix and S = eigenvector matrix of A. A must have n independent eigenvectors to make S invertible. All Ak = SA k SI.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

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.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

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

Hermitian matrix A H = AT = A.
Complex analog a j i = aU of a symmetric matrix.

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.

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

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).

Multiplicities AM and G M.
The algebraic multiplicity A M of A is the number of times A appears as a root of det(A  AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

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).

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

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Positive definite matrix A.
Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(DÂ» O.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.