midterm 1

Question 1

when are two systems of linear equations equivalent?

Answer 1

when their solution sets are the same

Question 2

a system of linear equations has:

Answer 2

1) no solution
2) exactly one solution
3) infinitely many solutions

Question 3

what are the two fundamental questions about a linear system?

Answer 3

1) is the system consistent?
2) if a solution exists, is it the only one?

Question 4

what are the conditions for a matrix to be in echelon form?

Answer 4

1) all nonzero rows are above any rows of zeros
2) each leading entry of a row is a column to the right of the leading entry in a row above it
3) all entries in a column below a leading entry are zero

Question 5

what are the conditions for a matrix to be in reduced row echelon form?

Answer 5

it satisfies all conditions to be in echelon form and:
1) the leading entry in each nonzero row is one
2) each leading one is the only nonzero entry in the column

Question 6

is the echelon form of a matrix unique?

Answer 6

no

Question 7

is the reduced echelon form of a matrix unique?

Answer 7

yes

Question 8

pivot position

Answer 8

the number and spot in the original matrix where a leading one is in the reduced echelon form of the same matrix

Question 9

pivot column

Answer 9

a column of a matrix that contains a pivot position

Question 10

when is a linear system consistent?

Answer 10

when it has one solution or infinitely many solutions

Question 11

when is a linear system inconsistent?

Answer 11

when it has no solution

Question 12

column vector

Answer 12

a matrix with only one column

Question 13

row vector

Answer 13

a matrix with only one row

Question 14

scalar multiple

Answer 14

cu, where u is a vector and c is a scalar

Question 15

u + v = ______

Answer 15

v + u

Question 16

(u + v) + w = ______

Answer 16

u + (v + w)

Question 17

u + 0 = ______

Answer 17

0 + u = u

Question 18

u + (-u) = _____

Answer 18

-u + u = 0

Question 19

c(u +v) = _____

Answer 19

cu + cv, where c is a scalar and u, v are vectors

Question 20

(c + d)u = ______

Answer 20

cu + du, where c, d are scalars and u is a vector

Question 21

c(du) = _____

Answer 21

(cd)u, where c, d are scalars and u is a vector

Question 22

1u = ____

Answer 22

u

Question 23

Span{v1, …, vp} is

Answer 23

the collection of all vectors that can be written in the form c1v1 + c2v2 + …. cpvp

Question 24

asking whether vector b is in Span{v1, …, vp} is the same as asking _________

Answer 24

if x1v1 + x2v2 + … + xpvp = b has a solution

Question 25

if matrix A has a pivot position in every row...

Answer 25

the columns of A span R^n

Question 26

a system of linear equations is homogeneous if:

Answer 26

it can be written in the form Ax = 0

Question 27

what is the trivial solution?

Answer 27

the vector x = 0 that makes Ax = 0

Question 28

what is a nontrivial solution?

Answer 28

when vector x does not equal zero and Ax = 0

Question 29

parametric vector form

Answer 29

x = su + tv

Question 30

two matrices are row equivalent if:

Answer 30

there exists a sequence of elementary row operations that transforms one matrix into the other

Question 31

a system of vectors is linearly independent if _________

Answer 31

the linear combination of the vectors x1v1 + x2v2 + ... + xpvp = 0 has only the trivial solution

Question 32

a system of vectors is linearly dependent if _________

Answer 32

if there exists weights such that c1v1 + c2v2 + ... + cpvp = 0, where at least 1 weight is nonzero

Question 33

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

Answer 33

linearly dependent

Question 34

if the set of vectors contains the zero vector, then the set is _______

Answer 34

linearly dependent

Question 35

if the set contains more vectors than there are entries in each vector, then the set is _______

Answer 35

linearly dependent

Question 36

a transformation T from IR^n to IR^m is ______

Answer 36

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

Question 37

domain of T

Answer 37

IR^n (original)

Question 38

codomain of T

Answer 38

IR^m (new)

Question 39

T(x)

Answer 39

the image of x

Question 40

range of T

Answer 40

the set of all images T(x)

Question 41

a transformation T is linear if:

Answer 41

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

Question 42

first part of transformation T being linear

Answer 42

T(u + v) = T(u) + T(v) for all u, v in the domain of T

Question 43

second part of transformation T being linear

Answer 43

T(cu) = cT(u) for all scalars c and all u in the domain of T

Question 44

a transformation T from IR^n to IR^m is ______

Answer 44

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

Question 45

in the equation Ax = b, what is A and what does A do?

Answer 45

A is a matrix, and it acts on or transforms vector x to produce a new vector b

Question 46

what does a sheer transformation look like?

Answer 46

it stretches the area of the matrix (the sheep is being pulled)

Question 47

dilation / contraction

Answer 47

given a scalar r, define T. IR^2 –> IR^2 by T(x) = rx (multiplying the values by a constant doesn’t change anything but the length of the arrow)

Question 48

dilation criteria

Answer 48

r > 1

Question 49

contraction criteria

Answer 49

0 <= r <= 1

Question 50

A(BC) =

Answer 50

(AB)C

Question 51

A(B + C) =

Answer 51

AB + AC

Question 52

(B + C)A =

Answer 52

BA + CA

Question 53

(A^T)^T =

Answer 53

A

Question 54

(A + B)^T =

Answer 54

A^T + B^T

Question 55

(rA^T) =

Answer 55

(A^T), r scalar

Question 56

(AB)^T =

Answer 56

B^T * A^T

Question 57

two matrices are equal if:

Answer 57

1) they are the same size 2) their columns are equal (all entries are equal)

Question 58

IA =

Answer 58

A

Question 59

A^3 =

Answer 59

A*A*A

Question 60

canonical linear programming problem

Answer 60

maximize f(x) = c^T * x subject to the constraints Ax <=b and x>= 0

Question 61

feasible solution

Answer 61

any vector x that satisfies the constraints Ax <= b and x>= 0

Question 62

feasible set

Answer 62

the set, F, of all feasible solutions

Question 63

slack variable

Answer 63

a nonnegative variable that is added to the smaller side of an inequality to convert it to an equality

Question 64

can a slack variable be negative?

Answer 64

NO

Question 65

when is F an empty set?

Answer 65

when the problem is infeasible

Question 66

when are there arbitrarily large values in F?

Answer 66

when the problem is unbounded

Question 67

unbounded

Answer 67

no maximum

Question 68

infeasible

Answer 68

no solution