MA 322

Question 1

linear equation

Answer 1

in the form

a1*x1 + a2*x2 … an*xn = b

a is constant; x is variable

Question 2

system of linear equations

(linear system)

Answer 2

collection of one or more

linear equations with

same variables

Question 3

solution set of

system of equations

Answer 3

list of number that

each make equation true

Question 4

equivalent linear systems

Answer 4

have same solution set

Question 5

consistent

Answer 5

has one or infinitely many solutions

Question 6

inconsistent

Answer 6

has no solutions

Question 7

coefficent matrix

Answer 7

matrix of coeffiecnts of linear system

Question 8

augmented matrix

Answer 8

coefficent matrix with constants from

right side of equation

Question 9

row equivalent

Answer 9

two matrices are row equivalent if

there is a sequence of elementary

row opertations that transforms one into

another

Question 10

existence

Answer 10

is the matrix consistent?

Question 11

uniqueness

Answer 11

is there only one solution?

Question 12

leading entry

Answer 12

left-most non-zero entry in a row

Question 13

echelon form

(row echelon form)

Answer 13

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

Question 14

reduced echelon form

(reduced row echelon form)

Answer 14

In echelon form:

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

Question 15

pivot position

Answer 15

location that corresponds to a leading one

in the reduced echelon form

Question 16

pivot column

Answer 16

column that contains a pivot position

Question 17

pivot

Answer 17

non-zero number in a pivot position

that is needed to create zeroes

in row operations

Question 18

basic variables

Answer 18

a variable in a linear system that

corresponds to a pivot column

in the coeffient matrix

Question 19

free variables

Answer 19

any variable that is not a

basic varible

Question 20

parametric description

Answer 20

uses free varibles as parameters

in form:

x1 = a +x3

x2 = b -x3

x3 is free

x4 = c

Question 21

vector

(column vector)

Answer 21

matrix with only one column

Question 22

linear combination

with vectors v1,v2, … vpinRⁿ

and weights (scalars) c1, c2, … cp

Answer 22

in form:

y = c1*v1 + c2*v2… + cp*vp

where y is the linear combination

Question 23

span

for v1, … vp in Rⁿ

Answer 23

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

Question 24

Span{u,v} in R³

Answer 24

the set of all scalar multiples of v

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

Question 25

Span{**u**,**v**} in **R**³

Answer 25

the plane that contains **u**, **v**, and **0**

Question 26

if A is an m X n matrix with columns **a1**... **an** and if **x** is in **R**ⁿ and the # of columns of A = # of entries in **x** then the product of A and **x** is:

Answer 26

the linear combination of the columns of A using the corresponding entries in **x** as weights A**x** = **b** 

Question 27

matrix equation

Answer 27

A**x** = **b** in form 

Question 28

the equation A**x** = **b** has a solution

Answer 28

if and only if **b** is a linear combination of the columns of A

Question 29

For A is an m X n matrix the following statements are either all true or all false (about coefficent matrix)

Answer 29

a. ) for each **b** in **R**^m the equation A**x** = **b** has a solution b. ) each **b** in **R**^m is a linear combination of the colmns of A c. ) the cloumns of A span **R**^m d. ) A has a pivot position in every row

Question 30

identity matrix (**I** or **I**_n)

Answer 30

a square matrix with ones on the diagonal and zeroes elsewhere (**I**_n \* **x** = **x** for all **x** in **R**ⁿ)

Question 31

if A is an m X n matrix **u** and **v** are vectors in **R**ⁿ and c is a scalar

Answer 31

a. ) A(**u** + **v**) = A**u** + A**v** b. ) A(c**u**) = c(A**u**)

Question 32

homogeneous

Answer 32

can be written in form A**x** = **0** always has one solution **x** = **0** (trival solution)

Question 33

the homogenous equation A**x** = **0** has a non-trival solution

Answer 33

if and only if the equation has at least one free variable

Question 34

in the form a1\*x1 + a2\*x2 ... an\*xn = b a is constant; x is variable

Answer 34

linear equation

Question 35

collection of one or more linear equations with same variables

Answer 35

system of linear equations (linear system)

Question 36

list of number that each make equation true

Answer 36

solution set of system of equations

Question 37

have same solution set

Answer 37

equivalent linear systems

Question 38

has one or infinitely many solutions

Answer 38

consistent

Question 39

has no solutions

Answer 39

inconsistent

Question 40

matrix of coeffiecnts of linear system

Answer 40

coefficent matrix

Question 41

coefficent matrix with constants from right side of equation

Answer 41

augmented matrix

Question 42

is the matrix consistent?

Answer 42

existence

Question 43

is there only one solution?

Answer 43

uniqueness

Question 44

left-most non-zero entry in a row

Answer 44

leading entry

Question 45

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 

Answer 45

echelon form (row echelon form)

Question 46

In echelon form: 1. the entry in each row is one 2. each leading entry is the only non-zero entry in its column 

Answer 46

reduced echelon form (reduced row echelon form)

Question 47

location that corresponds to a leading one in the reduced echelon form

Answer 47

pivot position

Question 48

column that contains a pivot position

Answer 48

pivot column

Question 49

non-zero number in a pivot position that is needed to create zeroes in row operations

Answer 49

pivot

Question 50

a variable in a linear system that corresponds to a pivot column in the coeffient matrix

Answer 50

basic variables

Question 51

any variable that is not a basic varible

Answer 51

free variables

Question 52

uses free varibles as parameters in form: x1 = a +x3 x2 = b -x3 x3 is free x4 = c

Answer 52

parametric description

Question 53

matrix with only one column

Answer 53

vector (column vector)

Question 54

in form: **y** = c1\***v1** + c2\***v2**... + cp\***vp** where **y** is the linear combination

Answer 54

linear combination with vectors **v****1**,**v2**, ... **vp**in**R**ⁿ and weights (scalars) c1, c2, ... cp

Question 55

the set of all linear combinations of **v**1 ... **vp** is denoted by Span{**v1**...**vp**} and is called the subset of **R**ⁿ spanned by **v1**... **vp**

Answer 55

span for **v1**, ... **vp** in **R**ⁿ

Question 56

the set of all scalar multiples of **v** (set of points on the line through **v** and **0**)

Answer 56

Span{**u**,**v**} in **R**³

Question 57

the plane that contains **u**, **v**, and **0**

Answer 57

Span{**u**,**v**} in **R**³

Question 58

the linear combination of the columns of A using the corresponding entries in **x** as weights A**x** = **b** 

Answer 58

if A is an m X n matrix with columns **a1**... **an** and if **x** is in **R**ⁿ and the # of columns of A = # of entries in **x** then the product of A and **x** is:

Question 59

A**x** = **b** in form 

Answer 59

matrix equation

Question 60

if and only if **b** is a linear combination of the columns of A

Answer 60

the equation A**x** = **b** has a solution

Question 61

a. ) for each **b** in **R**^m the equation A**x** = **b** has a solution b. ) each **b** in **R**^m is a linear combination of the colmns of A c. ) the cloumns of A span **R**^m d. ) A has a pivot position in every row

Answer 61

For A is an m X n matrix the following statements are either all true or all false (about coefficent matrix)

Question 62

a square matrix with ones on the diagonal and zeroes elsewhere (**I**_n \* **x** = **x** for all **x** in **R**ⁿ)

Answer 62

identity matrix (**I** or **I**_n)

Question 63

a. ) A(**u** + **v**) = A**u** + A**v** b. ) A(c**u**) = c(A**u**)

Answer 63

if A is an m X n matrix **u** and **v** are vectors in **R**ⁿ and c is a scalar

Question 64

can be written in form A**x** = **0** always has one solution **x** = **0** (trival solution)

Answer 64

homogeneous

Question 65

if and only if the equation has at least one free variable

Answer 65

the homogenous equation A**x** = **0** has a non-trival solution