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Applied Linear Algebra > midterm 1 > Flashcards

midterm 1 Flashcards

1.1, 1.2, 1.3, 1.4, graph and adjacency matrices, 1.5, 1.7, 1.8, 2.1, 9.2, & 9.3

1
Q

when are two systems of linear equations equivalent?

A

when their solution sets are the same

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2
Q

a system of linear equations has:

A

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

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3
Q

what are the two fundamental questions about a linear system?

A

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

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4
Q

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

A

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

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5
Q

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

A

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

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6
Q

is the echelon form of a matrix unique?

A

no

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7
Q

is the reduced echelon form of a matrix unique?

A

yes

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8
Q

pivot position

A

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

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9
Q

pivot column

A

a column of a matrix that contains a pivot position

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10
Q

when is a linear system consistent?

A

when it has one solution or infinitely many solutions

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11
Q

when is a linear system inconsistent?

A

when it has no solution

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12
Q

column vector

A

a matrix with only one column

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13
Q

row vector

A

a matrix with only one row

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14
Q

scalar multiple

A

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

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15
Q

u + v = ______

A

v + u

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16
Q

(u + v) + w = ______

A

u + (v + w)

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17
Q

u + 0 = ______

A

0 + u = u

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18
Q

u + (-u) = _____

A

-u + u = 0

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19
Q

c(u +v) = _____

A

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

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20
Q

(c + d)u = ______

A

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

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21
Q

c(du) = _____

A

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

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22
Q

1u = ____

A

u

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23
Q

Span{v1, …, vp} is

A

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

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24
Q

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

A

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

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25
Q

if matrix A has a pivot position in every row…

A

the columns of A span R^n

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26
Q

a system of linear equations is homogeneous if:

A

it can be written in the form Ax = 0

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27
Q

what is the trivial solution?

A

the vector x = 0 that makes Ax = 0

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28
Q

what is a nontrivial solution?

A

when vector x does not equal zero and
Ax = 0

29
Q

parametric vector form

A

x = su + tv

30
Q

two matrices are row equivalent if:

A

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

31
Q

a system of vectors is linearly independent if _________

A

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

32
Q

a system of vectors is linearly dependent if _________

A

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

33
Q

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

A

linearly dependent

34
Q

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

A

linearly dependent

35
Q

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

A

linearly dependent

36
Q

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

A

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

37
Q

domain of T

A

IR^n (original)

38
Q

codomain of T

A

IR^m (new)

39
Q

T(x)

A

the image of x

40
Q

range of T

A

the set of all images T(x)

41
Q

a transformation T is linear if:

A

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

42
Q

first part of transformation T being linear

A

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

43
Q

second part of transformation T being linear

A

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

44
Q

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

A

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

45
Q

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

A

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

46
Q

what does a sheer transformation look like?

A

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

47
Q

dilation / contraction

A

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)

48
Q

dilation criteria

A

r > 1

49
Q

contraction criteria

A

0 <= r <= 1

50
Q

A(BC) =

A

(AB)C

51
Q

A(B + C) =

A

AB + AC

52
Q

(B + C)A =

A

BA + CA

53
Q

(A^T)^T =

A

A

54
Q

(A + B)^T =

A

A^T + B^T

55
Q

(rA^T) =

A

(A^T), r scalar

56
Q

(AB)^T =

A

B^T * A^T

57
Q

two matrices are equal if:

A

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

58
Q

IA =

A

A

59
Q

A^3 =

A

AAA

60
Q

canonical linear programming problem

A

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

61
Q

feasible solution

A

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

62
Q

feasible set

A

the set, F, of all feasible solutions

63
Q

slack variable

A

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

64
Q

can a slack variable be negative?

A

NO

65
Q

when is F an empty set?

A

when the problem is infeasible

66
Q

when are there arbitrarily large values in F?

A

when the problem is unbounded

67
Q

unbounded

A

no maximum

68
Q

infeasible

A

no solution

Applied Linear Algebra (11 decks)

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