MATH 125 FINAL
linear equality
(i.e. −2x+5y=12) a mathematical expression in which all variables appear to the first power (linear) and which states an equality between two expressions (equality).
linear inequality
(i.e. −2x+5y≥12) is a linear expression containing an inequality sign rather than an equality sign. Graphically a linear inequality in x and y defines a half-plane, i.e. those points (x,y) for which the inequality is true (all points on one side or the other side of the associated line −2x+5y=12 ).
a line
1-parameter solution = _________
C) A bounded geometric region in the plane formed by three or more straight line segments intersection only at corners.
1.1 What is a polygon? A) A person who knows and is able to use several languages. B) Either a triangle or a rectangle. C) A bounded geometric region in the plane formed by three or more straight line segments intersection only at corners. D) An expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
C) (1,0)
1.1 What is the corner point of the feasibility region of the system of inequalities x+y<1 and x-y<1? A) (1,1) B) (1,2) C) (1,0) D) (0,1) E) (0,0)
C) objective function
1.2 A linear program is the problem of maximizing or minimizing a linear function subject to a system of linear inequality constraints. The function to be optimized is called the: A) subjective function B) constrained function C) objective function D) max-min function E) optimized function F) problematic function
D) y=-1+2x
1.2 Let f(x,y)=2x+y. Which of the following equations does NOT define a line of constancy for f? A) y=1-2x B) y=-2x C) y=-1-2x D) y=-1+2x
A) If the feasibility region is a polygon P, then the maximum can be found by evaluating the objective function at the corners of P and picking the maximum value.
1.2 Suppose that we want to maximize a linear function f over some feasibility region. Which of the following statements is correct? A) If the feasibility region is a polygon P, then the maximum can be found by evaluating the objective function at the corners of P and picking the maximum value. B) If the feasibility region is a polygon P, then the maximization problem has a solution, and the solution is located at the centroid of P. C) If the feasibility region is unbounded, then the maximization problem does not have any solution. D) If the feasibility region is a polygon, then the maximization problem does not have a solution.
Step 4 Solve the linear program using mathematical techniques. Step 1 Assign variables. Step 2 Define the objective function. Step 3 Define the constraints.
1.3 Outline the steps for solving a linear programming word problem _________ Solve the linear program using mathematical techniques. _________ Assign variables. _________ Define the objective function. _________ Define the constraints.
B) Using lines of constancy
1.3 Which method can be always used when dealing with an unbounded feasibility region? A) Placing additional restrictions B) Using lines of constancy C) Turning a maximization problem into a minimization problem or vice versa D) Renaming the variables
B) Finding the inverse of the objective function.
1.3 Which of the following steps is NOT useful for solving a linear programming word problem? A) Graphing the feasibility region. B) Finding the inverse of the objective function. C) Evaluating the objective function at the corners. D) Finding the corner points.
a plane
2-parameter solution = ____________
2) coefficient matrix & 6) augmented matrix
2.1 A matrix is a rectangular array of numbers arranged in rows and columns. Matrices are used (among others) to compactly write and work with systems of linear equations. A system of linear equations has two important matrices associated with it. Select the correct two. 1) square matrix 2) coefficient matrix 3) system matrix 4) linear matrix 5) transformation matrix 6) augmented matrix
3) Multiply one row by another row.
2.1 We can perform a series of Elementary Row Operations and the solution of the corresponding linear system does not change. Which of the following four operations is NOT an elementary row operation? 1) Interchange two rows. 2) Multiply a row by a non-zero number. 3) Multiply one row by another row. 4) Add a multiple of one row to another row.
1) Gaussian elimination
2.1 We can solve a linear system of equations using the following four steps: Step 1: Rewrite linear system as a matrix. Step 2: Perform elementary row operations. Step 3: Obtain matrix in row echelon form (REF). Step 4: Perform back substitution. The procedure for using elementary row operations to move from a matrix to its row echelon form is called: 1) Gaussian elimination 2) Row echelonization 3) Back substitution 4) Elementary row manipulation 5) Echelon algorithm
C) it has one or more solutions
2.2 A system of linear equations is called consistent if A) the Gauss-Jordan elimination consistently gives the same answer B) it is consistent with its augmented matrix C) it has one or more solutions D) it has exactly one solution
D) two parallel lines
2.2 Geometrically, an inconsistent system of two equations in two variables x and y corresponds to: A) two intersecting lines B) two perpendicular lines C) two overlapping lines D) two parallel lines
D) the number of non-zero rows in the matrixs row echelon form
2.2 The row rank of a matrix is A) the number of rows in the matrix B) the number of non-zero rows in the matrix C) the number of non-zero entries in the first row of the matrix D) the number of non-zero rows in the matrixs row echelon form E) the number of non-zero entries in the matrix
D) are 0
2.3 An homogeneous system is a system of linear equations where all right-hand sides A) are equal B) are non-trivial C) are positive D) are 0 E) are consistent
D) first octant
2.3 The portion of the Euclidean 3-space where all x, y, and z are non-negative is called: A) first half B) first tri-sector. C) first quarter D) first octant E) first quadrant F) first sextant
D) a parametric solution
2.3 The solution of a linear system that has an infinite number of solutions is called: A) an infinite solution B) a boundless solution C) an incalculable solution D) a parametric solution E) a multisystem solution
B) a plane
2.4 Geometrically, the points (x,y,z) that satisfy 5x+4y-7z=2 define: A) a line B) a plane C) a point D) the whole Euclidean 3-space
D) one point
2.4 In general, a linear equation in three variables x,y,z has the form Ax+By+Cz=D (where A,B,C,D are some constants) and geometrically corresponds to a plane in 3D. Graphically, the solution of a system of two linear equations in three variables CANNOT be: A) a line B) a plane C) empty D) one point
A) choosing a non-zero entry in a matrix, then using row operations to turn it into a 1 and to create zeros above and below it.
2.4 Pivoting is the procedure of A) choosing a non-zero entry in a matrix, then using row operations to turn it into a 1 and to create zeros above and below it. B) choosing a non-zero entry in a matrix, then using row operations to turn it into a 1 and to create positive numbers above and zeros below it. C) choosing a non-zero entry in a matrix, then using row operations to create zeros above and below it. D) voting for the best Pi costume
A) The number of constraints is not greater than the number of variables.
2.5 The Simplex Algorithm is an algebraic method for finding a maximum in a linear program with two or more variables. We can only use the Simplex Algorithm if our linear program satisfies three of these requirements. Which property is NOT required? A) The number of constraints is not greater than the number of variables. B) The constraints need to have the form 'linear expression ≤ a positive number'. C) All variables must be non-negative. D) The linear program must be a maximization.
B) introducing slack variables
2.5 The initial simplex table is the augmented matrix required for the simplex algorithm. Before writing down the initial simplex table, we need to convert inequalities into equations. We do so by A) adding large positive constants B) introducing slack variables C) simply switching inequality signs into equality signs D) choosing pivots and performing pivoting
B) the vector starting at the origin and ending at the point P
3.1 The position vector of a point P is A) the vector starting at at the point P and ending at the origin B) the vector starting at the origin and ending at the point P C) the vector starting at the point P and ending at the point P
C) 4/3
3.1 What is the length of the vector u=(1/3, -1, 2/3, 0, 1/3, -1/3)? A) 2/9 B) 2/3 C) 4/3 D) 4/9 E) 16/9
B) position form
3.1 Which of the following forms is NOT used to describe a line? A) parametric form B) position form C) vector equation form D) standard form
A) if we increase the angle between u and v, the dot product u · v decreases.
3.2 The dot product can be defined by u · v = |u| · |v| · cos(α), where α is the angle between u and v. It follows that: A) if we increase the angle between u and v, the dot product u · v decreases. B) if we increase the angle between u and v, the dot product u · v stays the same. C) if we increase the angle between u and v, the dot product u · v increases.
C) 4
3.2 The dot product of the two vectors (1,7,-2) and (8,0,2) is equal to A) (9,7,0) B) 12 C) 4 D) (1,7,-2,8,0,2) E) (8,0,-4)
A) 0
3.2 Two vectors are orthogonal (=perpendicular) if their dot product is equal to A) 0 B) -1 C) 1 D) 1/2 E) 90°
C) The RHS column b can be written as a linear combination of the columns of the coefficient matrix A.
3.3 A system with augmented matrix [A|b] is consistent if and only if A) The RHS column b cannot be written as a linear combination of the rows of the coefficient matrix A. B) The RHS column b can be written as a linear combination of the rows of the coefficient matrix A. C) The RHS column b can be written as a linear combination of the columns of the coefficient matrix A. D) The RHS column b cannot be written as a linear combination of the columns of the coefficient matrix A.
C) (1,2,4)
3.3 Which vector CANNOT be written as a linear combination of the vectors (1,2,3) and (4,0,1)? A) (5,2,4) B) (3,-2,-2) C) (1,2,4) D) (2,4,6) E) (-2,4,5)
E) undefined
4.1 The product of two 2x3 matrices A(sub)(2x3) and B(sub)(2x3) is A) a 4x3 matrix B) a 2x9 matrix C) a 4x9 matrix D) a number E) undefined F) a 2x3 matrix
C) a 2x3 matrix
4.1 The sum of two 2x3 matrices A(sub)(2x3) and B(sub)(2x3) is A) a 2x6 matrix B) a 4x3 matrix C) a 2x3 matrix D) a 4x6 matrix E) undefined F) a number
B) A*B=I
4.2 A matrix B is the inverse of a matrix A if A) A*B=A B) A*B=I C) B+0=0 D) A*B=B E) A+B=A
A) x=A^(-1)*b
4.2 Inverses of matrices are used for solving systems of linear equations. Suppose that we rewrite the linear system as A*x=b, where A is an invertible matrix and x, b are vectors. Then we can express the solution x as: A) x=A^(-1)*b B) x=b*A C) x=b*A^(-1) D) x=A*b
A) A must be a square matrix and the RREF of A must be the identity matrix.
4.2 Which of the following conditions is sufficient for a matrix A to be invertible? A) A must be a square matrix and the RREF of A must be the identity matrix. B) The RREF of A must be the identity matrix. C) A must be a square matrix. D) The REF of A must be invertible and the RREF of A must be the identity matrix.
B) find the inverse of the matrix I-C (if it exists) and multiply it by d
4.3 Finally, to obtain the production vector x, we A) find the inverse of the matrix C (if it exists) and multiply it by d B) find the inverse of the matrix I-C (if it exists) and multiply it by d C) multiply the matrix I+C by d find the inverse of the matrix I+C (if it exists) and D) multiply it by d E) multiply the matrix I-C by d
B) depends only on the state of the system at the preceding time.
4.4 A Markov process is a process in which the probability of a system being in a particular state at a specific time A) is smaller than the probability of a system being in a particular state at the preceding time. B) depends only on the state of the system at the preceding time. C) is determined by an increasing sequence of probabilities. D) is a number between 0 and 1, found as a product of all previous outcomes.
Markov chain; Markov process
A __________ or a ____________ is a process in which the probability of a system being in a particular state at a specific time depends only on the state of the system in the preceding time period
consumption matrix
A _____________ is a matrix that represents the cost per dollar to run several companies or industries in an economy
transition matrix
A ________________ is a matrix whose entries represent the probabilities of moving from one state to another.
slack variable
A ________________ is an extra variable that is added to an inequality to turn the inequality into an equality, i.e. 5x(sub)1+x(sub)2≤9 ⟶ 5x(sub)1+x(sub)2+x(sub)3=9
stochastic matrix/vector
A _________________ is one where all the entries are non-negative and every column sums to 1
homogeneous solution
A _________________________ is one where all right-hand sides are 0. In the corresponding augmented matrix of the system, this means that the final column is all zeros.
D) All the above
A dot product is A) commutative B) distributive C) associative D) All the above
reduced row echelon form (RREF)
A matrix is in _____________________ if it is in row echelon form and has the additional property that above each leading 1 are only zeros.
1) 1 2) 0 3) right 4) bottom
A matrix is in row echelon form (REF) if it has the following four properties: 1) the first non-zero entry in each non-zero row is a ___ (the leading 1 in that row); 2) all entries directly below a leading 1 are ____'s; 3) as you move down the rows, the leading 1's move to the _______; and 4) any zero row is at the _________ (there could be several zero rows).
A) u+v B) cu
A non-empty subset V of vectors in Rn is a subspace of Rn if: A) whenever u and v are in V, then ______ is in V (closure under vector addition) AND B) whenever u is in V, then _____ is in V for any real number c (closure under multiplication by real numbers)
stable vector
A stochastic vector S associated with a transition matrix T is a ________________ if TS=S . (The vector S is called the equilibrium vector of the model.)
consistent
A system Ax=b is _______ if and only if b is in the column space of A.
augmented matrix
A system is consistent if and only if (iff) the row rank of the coefficient matrix equals the row rank of the ______________________.
consistent
A system of linear equations is _____________ if it has one or more solutions
consistent
A system with augmented matrix [A|b] is ________ if and only if b can be written as a linear combination of the columns of the coefficient matrix A .
linear combination
A vector v is called a _______________ of v(sub)1,v(sub)2,...,v(sub)k if v=c(sub)1v(sub)1+c(sub)2v(sub)2+⋯+c(sub)kv(sub)k for some scalars (i.e. real numbers) c(sub)1, c(sub)2 ,..., c(sub)k .
1) less than 2) less than 3) positive
An economy is productive if ... 1) every column of C sums to _______ 1 (i.e. every industry is profitable); OR 2) every row of C sums to __________ 1 (i.e. industries do not consume all the resources in production and so there will be leftovers to sell); OR 3) (I−C)^(−1) exists and has all ________ entries.
subspace
Any plane that does not pass through the origin cannot be a __________
position vector
Any point P in Euclidean space Rn can be thought of as a vector starting at the origin O and ending at the point. This is the _____________ of the point P, denoted P
equation of a plane in standard form
Ax+By+Cz=D where A, B, C, and D are constants.
Unique solution = a point
Consistent, no free variables
1-parameter solution = a line
Consistent, one free variable
2-parameter solution = a plane
Consistent, two free variables
zero
Every subspace must contain the _____ vector O of Rn
1) # of variables 2) infinitely many solutions
For an homogeneous system 1) if the row rank of the coefficient matrix equals the _________________, then the system has only the trivial solution. 2) if the row rank of the coefficient matrix is less than the number of variables, then the system has ____________________ —that is, some free variables are present and so some parameters appear in the solution.
consistent
Homogeneous systems are always _________________
C) We can run facility B for 3 days and facility C for one day to make exactly the same amounts of products as facility A would make in one day.
In applications, it is advantageous if we can express a vector as a linear combination of other vectors because that gives us more freedom on how to fulfill orders, substitute in recipes, ... For example, suppose that we have three production facilities A, B, C, where each facility makes different amounts of the same products per day. Denote the corresponding production vectors by A, B, C. If A=3B+C, then we know that: A) We can run facility B for 3 days and facility C for one day to make more products than facility A would make in one day. B) It is cheaper to run facility A instead of facilities B and C. C) We can run facility B for 3 days and facility C for one day to make exactly the same amounts of products as facility A would make in one day. D) The cost to run facility A per one day is the same as running facility B for 3 days and facility C for one day.
maximum
In situations when the simplex table still has a pivot column but no candidates for pivoting (no positive entries in the pivot column), there is no ____________ for the linear program.
line of constancy
Let f be a linear function: f(x,y)=ax+by. A _________________ for f is a line along which f(x,y) has a constant value, i.e. ax+by=C for some constant C.
square
Only ________ matrices have inverses, but not every _______ matrix has an inverse (same term twice)
-Graph the constraints (the feasibility region). -Find the corner points. -Evaluate the objective function z at the corner points. -Give the location and values of the max/min of z
Procedure for solving a linear program with a polygonal feasibility region
F; there can only be one inverse if there is one
T or F: if there is an inverse there can be multiple inverses
True
T or F: vector addition is commutative
1) interchangeable 2) multiply 3) add
The Three Elementary Row Operations 1) __________ two rows. Ri↔Rj (i.e. R′i=Rj and R′j=Ri) 2) _________ a row by a non-zero number. R′i=kRi 3) _____ a multiple of one row to another. R′j=Rj+kRi
null space
The _______ of a matrix A , denoted nul (A), is the set of all solutions of the homogeneous system Ax=O.
column space
The _________ of a matrix A , denoted col(A) , is the set of all linear combinations of the columns of A. If A=[a(sub)1 a(sub)2 ... a(sub)n] , then col(A)=span{a(sub)1,a(sub)2,...,a(sub)n} .
scalar multiple
The _____________ of the matrix A by the scalar k is the matrix kA defined by kA=[kaij]
row rank
The ______________ of a matrix is the number of non-zero rows in the matrix's row echelon form.
dot product
The _______________ u⋅v is defined to be the real number u⋅v = u(sub)1v(sub)1+u(sub)2v(sub)2+⋯+u(sub)nv(sub)n
basic feasible solution
The ___________________ of a simplex table is a point in the space of original and slack variables where z takes on the value in the upper right corner of the table.
parametric solution
The solution of a linear system that has an infinite number of solutions is called a _________________ since it expresses the variables in terms of a collection of parameters. (Other terms for these parameters are free variables or independent variables.)
0
The trivial solution of a homogeneous system is the solution where all variables are equal to ____. All other solutions, if there are any, are called non-trivial
length
The________, or magnitude, or norm, or modulus, of the vector u=[u(sub)1,u(sub)2,...,u(sub)n] is |u|=√(u(sub)1^2+u(sub)2^2+⋯+u(sub)n^2).
describe a line
Three Ways to _________________ Mathematically: 1) standard form: y=mx+b (though only on the plane R2) 2) parametric form 3)vector equation form: x=P0+tv for all t∈R where x=(x(sub)1,x(sub)2,...,x(sub)n), P(sub)0=(p(sub)1,p(sub)2,...,p(sub)n), and v=(v(sub)1,v(sub)2,...,v(sub)n)
dimensions
Two matrices are equal if they have the same ________ and their corresponding entries are equal to each other.
parallel
Two non-zero vectors u and v are __________ if one is a scalar multiple of the other: u=kv, where k≠0 .
Rn
V=____ is the largest subspace
O
V={__} is the smallest subspace.
orthogonal
Vectors u=(u(sub)1,u(sub)2,...,u(sub)n) and v=(v(sub)1,v(sub)2,...,v(sub)n) in R^n are _____________, written u⊥v, if u⋅v=0.
x=P0+tv
We can write the vector equation of a line _______ in the new form x=P0+ span{v}
Gaussian elimination
a systematic procedure for using elementary row operations to move from a matrix to its row echelon form (and hence without changing the set of solutions).
linear function
any function in which every variable in the expression defining the function is raised to the first power (i.e. f(x,y)=2x−(5/7)y AND/OR g(x,y)=7x+8y , etc.).
1) objective function 2) slack variables
initial simplex table is the augmented matrix required for the simplex algorithm and is formed from the linear system that arises when: 1) the _______________ is written in standard form (as a linear homogeneous equation); and 2) all inequalities are converted to equations by introducing _________________
perpendicular
orthogonal means
line
row rank is 1
plane
row rank is 2
all of R^3
row rank is 3
x=P(sub)0+su+tv x=general point in the plane P(sub)0= specific point on the plane u & v = any two vectors that are not parallel
vector equation of a plane has the form _______ with s,t in R
x=P(sub)0+span {u,v}
what is another way to write x=P(sub)0+su+tv