finals: sections 1.5 - 9.2

Question 1

when is a system of linear equations considered to be homogeneous?

Answer 1

when it can be written in the form Ax = 0

Question 2

the homogeneous equation Ax = 0 has a nontrivial solution if and only if _____________

Answer 2

the equation has at least one free variable

Question 3

what is the trivial solution?

Answer 3

the equation Ax = 0, where x = 0 is the only solution

Question 4

what is a nontrivial solution?

Answer 4

an x does not equal 0 that satisfies Ax = 0

Question 5

what is parametric vector form?

Answer 5

when the solution is written as a vector equation, x = su +tv

Question 6

what is linear independence?

Answer 6

a set {v1, …, vp} is linearly independent if the vector equation x1v1 + … + xpvp = 0 has only the trivial solution

Question 7

the columns of a matrix A are linearly independent if and only if _______________

Answer 7

the equation Ax = 0 has only the trivial solution

Question 8

if a set has more vectors than entries in each vector, then the set is ___________

Answer 8

linearly dependent

Question 9

if a set of vectors contains the zero vector, then the set is _________

Answer 9

linearly dependent

Question 10

when two vectors are multiples of each other, they are _______________

Answer 10

linearly dependent

Question 11

why do we care about linear independence?

Answer 11

1) it helps us understand whether a set of vectors provides unique information
2) if the set is linearly dependent, some of the vectors are redundant because they can be written as a combination of other vectors

Question 12

a transformation or mapping, T, is linear if:

Answer 12

1) T(u+v) = T(u) + T(v), for all vectors u and v in the domain of T
2) T(cu) = cT(u), for all scalars c and all vectors u in the domain of T

Question 13

what is a transformation (or mapping) T from IR^n to IR^m?

Answer 13

a rule that assigns a vector x in IR^n to a vector T(x) in IR^m

Question 14

what is the domain of T?

Answer 14

IR^n

Question 15

what is the codomain of T?

Answer 15

IR^m

Question 16

what is the image of x?

Answer 16

T(x)

Question 17

what is the range of T?

Answer 17

the set of all images T(x)

Question 18

what is a shear transformation?

Answer 18

the transformation from IR^2 –> IR^2 defined by T(x) = Ax

Question 19

what is a dilation?

Answer 19

given scalar r, the transformation from IR^2 –> IR^2 defined by T(x) = rx

Question 20

what is a diagonal matrix?

Answer 20

a square, nxn matrix with entries on the diagonal and zeros everywhere else

Question 21

what are the properties of matrices and matrix multiplication?

Answer 21

1) A(BC) = (AB)C
2) A(B + C) + AB + AC
3) (B + C)A + BA + CA

Question 22

what is the transpose of a matrix?

Answer 22

given an nxm matrix A, the transpose of A, denoted A^T, is the mxn matrix whose columns are formed from the corresponding rows of A

Question 23

what are the properties of transposing?

Answer 23

1) (A^T)^T = A
2) (A + B)^T = A^T + B^T
3) (rA^T) = r(A^T), r scalar
4) (AB)^T + B^T * A^T

Question 24

what is the geometric approach good for?

Answer 24

lower dimensional problems (3D or less)

Question 25

what is the canonical linear programming problem?

Answer 25

1) maximize c^Tx = f(x) subject to:
2) the constraints of Ax <= b
3) and x > 0

Question 26

what is an optimal vector?

Answer 26

a vector that satisfies 1, 2, and 3

Question 27

what is a feasible solution?

Answer 27

a vector that satisfies 2 and 3

Question 28

what is an objective function?

Answer 28

what you are optimizing

Question 29

what are the two things that can go wrong in a linear programming problem?

Answer 29

1) infeasible
2) unbounded

Question 30

what is infeasible?

Answer 30

the constraint inequalities are inconsistent and the set of all feasible solutions F, is an empty set

Question 31

what is unbounded?

Answer 31

there are arbitrarily large values in F, then a maximum DNE