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Matrix Algebra > MA 322 > Flashcards

MA 322 Flashcards

1
Q

linear equation

A

in the form

a1*x1 + a2*x2 … an*xn = b

a is constant; x is variable

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2
Q

system of linear equations

(linear system)

A

collection of one or more

linear equations with

same variables

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3
Q

solution set of

system of equations

A

list of number that

each make equation true

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4
Q

equivalent linear systems

A

have same solution set

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5
Q

consistent

A

has one or infinitely many solutions

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6
Q

inconsistent

A

has no solutions

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7
Q

coefficent matrix

A

matrix of coeffiecnts of linear system

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8
Q

augmented matrix

A

coefficent matrix with constants from

right side of equation

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9
Q

row equivalent

A

two matrices are row equivalent if

there is a sequence of elementary

row opertations that transforms one into

another

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10
Q

existence

A

is the matrix consistent?

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11
Q

uniqueness

A

is there only one solution?

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12
Q

leading entry

A

left-most non-zero entry in a row

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13
Q

echelon form

(row echelon form)

A
  1. All non-zero rows are above any rows of all zeroes
  2. Each leading entry of a row is in a column to the right of the leading entry of the row above it
  3. All entries in a column below a leading entry are zero
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14
Q

reduced echelon form

(reduced row echelon form)

A

In echelon form:

  1. the entry in each row is one
  2. each leading entry is the only non-zero entry in its column
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15
Q

pivot position

A

location that corresponds to a leading one

in the reduced echelon form

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16
Q

pivot column

A

column that contains a pivot position

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17
Q

pivot

A

non-zero number in a pivot position

that is needed to create zeroes

in row operations

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18
Q

basic variables

A

a variable in a linear system that

corresponds to a pivot column

in the coeffient matrix

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19
Q

free variables

A

any variable that is not a

basic varible

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20
Q

parametric description

A

uses free varibles as parameters

in form:

x1 = a +x3

x2 = b -x3

x3 is free

x4 = c

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21
Q

vector

(column vector)

A

matrix with only one column

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22
Q

linear combination

with vectors v1,v2, … vpinRn

and weights (scalars) c1, c2, … cp

A

in form:

y = c1*v1 + c2*v2… + cp*vp

where y is the linear combination

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23
Q

span

for v1, … vp in Rn

A

the set of all linear combinations

of v1 … vp is denoted by Span{v1…vp} and is called

the subset of Rn spanned by

v1… vp

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24
Q

Span{u,v} in R3

A

the set of all scalar multiples of v

(set of points on the line through v and 0)

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25
Q

Span{u,v} in R3

A

the plane that contains u, v, and 0

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26
Q

if A is an m X n matrix

with columns a1… an and if x is in Rn

and the # of columns of A = # of entries in x

then the product of A and x is:

A

the linear combination of the columns of A using the

corresponding entries in x as weights

Ax = b

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27
Q

matrix equation

A

Ax = b

in form

28
Q

the equation Ax = b

has a solution

A

if and only if

b is a linear combination

of the columns of A

29
Q

For A is an m X n matrix

the following statements are either

all true or all false

(about coefficent matrix)

A

a. ) for each b in Rm the equation Ax = b has a solution
b. ) each b in Rm is a linear combination of the colmns of A
c. ) the cloumns of A span Rm
d. ) A has a pivot position in every row

30
Q

identity matrix

(I or In)

A

a square matrix with ones on the diagonal

and zeroes elsewhere

(In * x = x for all x in Rn)

31
Q

if A is an m X n matrix

u and v are vectors in Rn

and c is a scalar

A

a. ) A(u + v) = Au + Av
b. ) A(cu) = c(Au)

32
Q

homogeneous

A

can be written in form Ax = 0

always has one solution

x = 0

(trival solution)

33
Q

the homogenous equation

Ax = 0 has a non-trival solution

A

if and only if

the equation has at least one free variable

34
Q

in the form

a1*x1 + a2*x2 … an*xn = b

a is constant; x is variable

A

linear equation

35
Q

collection of one or more

linear equations with

same variables

A

system of linear equations

(linear system)

36
Q

list of number that

each make equation true

A

solution set of

system of equations

37
Q

have same solution set

A

equivalent linear systems

38
Q

has one or infinitely many solutions

A

consistent

39
Q

has no solutions

A

inconsistent

40
Q

matrix of coeffiecnts of linear system

A

coefficent matrix

41
Q

coefficent matrix with constants from

right side of equation

A

augmented matrix

42
Q

is the matrix consistent?

A

existence

43
Q

is there only one solution?

A

uniqueness

44
Q

left-most non-zero entry in a row

A

leading entry

45
Q
  1. All non-zero rows are above any rows of all zeroes
  2. Each leading entry of a row is in a column to the right of the leading entry of the row above it
  3. All entries in a column below a leading entry are zero
A

echelon form

(row echelon form)

46
Q

In echelon form:

  1. the entry in each row is one
  2. each leading entry is the only non-zero entry in its column
A

reduced echelon form

(reduced row echelon form)

47
Q

location that corresponds to a leading one

in the reduced echelon form

A

pivot position

48
Q

column that contains a pivot position

A

pivot column

49
Q

non-zero number in a pivot position

that is needed to create zeroes

in row operations

A

pivot

50
Q

a variable in a linear system that

corresponds to a pivot column

in the coeffient matrix

A

basic variables

51
Q

any variable that is not a

basic varible

A

free variables

52
Q

uses free varibles as parameters

in form:

x1 = a +x3

x2 = b -x3

x3 is free

x4 = c

A

parametric description

53
Q

matrix with only one column

A

vector

(column vector)

54
Q

in form:

y = c1*v1 + c2*v2… + cp*vp

where y is the linear combination

A

linear combination

with vectors v1,v2, … vpinRn

and weights (scalars) c1, c2, … cp

55
Q

the set of all linear combinations

of v1 … vp is denoted by Span{v1…vp} and is called

the subset of Rn spanned by

v1… vp

A

span

for v1, … vp in Rn

56
Q

the set of all scalar multiples of v

(set of points on the line through v and 0)

A

Span{u,v} in R3

57
Q

the plane that contains u, v, and 0

A

Span{u,v} in R3

58
Q

the linear combination of the columns of A using the

corresponding entries in x as weights

Ax = b

A

if A is an m X n matrix

with columns a1… an and if x is in Rn

and the # of columns of A = # of entries in x

then the product of A and x is:

59
Q

Ax = b

in form

A

matrix equation

60
Q

if and only if

b is a linear combination

of the columns of A

A

the equation Ax = b

has a solution

61
Q

a. ) for each b in Rm the equation Ax = b has a solution
b. ) each b in Rm is a linear combination of the colmns of A
c. ) the cloumns of A span Rm
d. ) A has a pivot position in every row

A

For A is an m X n matrix

the following statements are either

all true or all false

(about coefficent matrix)

62
Q

a square matrix with ones on the diagonal

and zeroes elsewhere

(In * x = x for all x in Rn)

A

identity matrix

(I or In)

63
Q

a. ) A(u + v) = Au + Av
b. ) A(cu) = c(Au)

A

if A is an m X n matrix

u and v are vectors in Rn

and c is a scalar

64
Q

can be written in form Ax = 0

always has one solution

x = 0

(trival solution)

A

homogeneous

65
Q

if and only if

the equation has at least one free variable

A

the homogenous equation

Ax = 0 has a non-trival solution

Matrix Algebra (1 decks)

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