Math Quiz #1

Front 1

row equivalence

Back 1

sequence of simple row operations that make one matrix into other

Front 2

row equivalence and augmented matrices

Back 2

if two augmented matrices are row equivalent, then two systems have same solution set

Front 3

what makes something in echelon form?

Back 3

zeros are at the bottom of the matrix, not at the top
each leading entry of a row is to the right of the one above it
all entries in a column below a leading entry are zeros

Front 4

RREF classification?

Back 4

leading entry must be 1
leading entry is the only nonzero number in that column

Front 5

what is a leading entry?

Back 5

leftmost nonzero entry

Front 6

pivot position properties vs. pivot properties

Back 6

in RREF, pivot positions are fixed no matter what multi or add you do

pivot is the actual number in the column, can be many different numbers

Front 7

basic variables

Back 7

variables corresponding to pivot columns

Front 8

general solution/parametric descriptions form

Back 8

x1=gsdgs
x2=3sgfg
x3 is free

thus, it gives explicit descriptions of all solutions

Front 9

parametric description

Back 9

whenever a solution set has free variables/consistent, it has many parametric descriptions

Front 10

how to write basic variables in general solution?

Back 10

write basic variables in terms of free variables

Front 11

vector

Back 11

a list of numbers, a one column matrix

Front 12

[w1]
[w2] = w

what are w1 and w2?

Back 12

ordered pairs

Front 13

vectors = vectors iff

Back 13

corresponding entries are equal
7 4
4 does not equal 7

Front 14

R^n

Back 14

n= how many rows
R is real numbers that appear as entries in the vectors

Front 15

R^2

Back 15

2x1 column matric

Front 16

R^3

Back 16

3x1 column matrix

Front 17

zero vector

Back 17

a vector whose entries are all zero, represent one single point

Front 18

q: determine whether b is a linear combo of a1 a2

Back 18

x1a1 +x2a2 = b

so row reduce the augmented matrix to see if it is consistent. if consistent, than b is a linear combo of a1a2

Front 19

the span of {v1...vp} is the set

Back 19

of all vectors linear combinations

Front 20

q: is b is in the span{v1..vp}

Back 20

is b a linear combination??

thus x1v1 +x2v2 = b?

Front 21

geometrically defining span

Back 21

the line/plane with all the possible linear combinations

Front 22

zero vector & span

Back 22

0 vector must be always be in span

Front 23

AX=B

Back 23

A is an m*n matrix with all the vectors
X representes weights
B is the product of A and X, assuming x is in R^n

Front 24

vector equation

Back 24

x1a1 +x2a2 +x3a3 =b

Front 25

matrix equation

Back 25

[a1 a2 a3][x1]
[x2]
[x3]

Front 26

statements that are logically equivalent

Back 26

for each b in R^m the equation AX=B has a solution
each B in R^m is a linear combo of colums of A
the columns of A span R^m
A has a pivot position in every row

Front 27

homogenous system

Back 27

AX=0

Front 28

trivial solution

Back 28

when x=0 is a solution

Front 29

non trivial solution

Back 29

for AX=0, nonzero vector x that satisfies the equation
AX=0 will only have a nontrivial solution if and only if the equation has at least one free variable

Front 30

how to write solution set for homogeneous systems

Back 30

x=[], then pull out x1 or x3 so you have

xv

Front 31

how to describe solution set for a homogeneous equation

Back 31

span {v1..vp}

Front 32

how to write a solution to AX=B in parametric vector form

Back 32

x2[] +x3[] = x

Front 33

translation

Back 33

vector additon

Front 34

definition of translation geometrically

Back 34

moves vector x in a parallel direction by adding p

Front 35

if a set contains a zero vector

Back 35

linearly dependent

Front 36

linear independence

Back 36

no free variables

columns of matrix A are linearly independent

if Ax=0 has only the trivial solution

SETS OF ONE/TWO VECTORS: if a set containing one vector is not a zero vector

SETS OF TWO OR MORE VECTORS:
rows>columns don't mean shit

Front 37

linearly dependence

Back 37

free variables that don't give solutions that are all zeros

an indexed set S={v1...vp} is linearly dependent iff at least one vector in S is a linear combo of others

SETS OF ONE OR TWO VECTORS
if v1 is a multiple of v2, sets are linearly dependent

if a set contains a 0 vector

SETS OF TWO OR MORE VECTORS:
if columns > rows (more variables than equations, so there must be a free variable)

Front 38

T:R^n to R^m

Back 38

transformation, function, mapping

assigns to each vector x in R^n a vector t(x) in R^m

Front 39

R^n in transformations

Back 39

domain of T

Front 40

R^m in transformations

Back 40

codomain

Front 41

image of x

Back 41

for x in R^n, the vector T(x) in R^m

Front 42

range

Back 42

the set of all images T(x) is called range

set off all linear combinations of columns of A because T(x) = Ax

Front 43

matrix transformations

Back 43

T(x) behaves like Ax

Front 44

Find T(u) the image of u under the transformation

Back 44

Compute T(u) which is Au

Front 45

Find an x in R^2 whose image under T is b

Back 45

Solve T(x)=b for x, or solve Ax= b

Front 46

Is there more than one x whose image under T is b?

Back 46

Is the system unique or not unique?

Front 47

Determine if c is in the range of the trans T

Back 47

Is T(x)=c consistent?

Front 48

shear transformation

Back 48

produces parallelogram

Front 49

A transformation is linear if

Back 49

T(u+v) = Tu + Tv
T(cu) = cT(u)

Front 50

matrix transformations =

Back 50

linear transformation

Front 51

horizontal contraction/dialation

Back 51

k is in upper left

Front 52

vertical contraction/dialation

Back 52

k is in lower right (diagonal to horizontal contraction)

Front 53

horizontal shear

Back 53

k is in the upper right hand

Front 54

vertical shear

Back 54

k is in the lower left hand

Front 55

onto

Back 55

T maps if and only if columns
existence question
span R^m or every vector in R^m is line

no zero row in REF of A

Front 56

one to one

Back 56

uniqueness question

pivot in each column
has a unique solution or none at all

Front 57

not onto

Back 57

existence question
when there is no solution

Front 58

not one to one

Back 58

uniqueness question
more than one solution

Front 59

scalar multiple

Back 59

rA is a scalar multiple of A when r is a scalar

Front 60

commute

Back 60

if AB = BA

Front 61

AB= AC, B does not equal C

Back 61

True

Front 62

transpose of A is:

Back 62

n*m matrix when A is m*n
denoted by ^T

Front 63

(A^T)^T =

Back 63

A

Front 64

(AB)^T

Back 64

B^T * A^T

Front 65

invertible matrix A and C theorem

Back 65

both must be square and when you multiply them together, it equals I

Front 66

singular matrix

Back 66

matrix that is not invertible

Front 67

nonsingular matrix

Back 67

invertible matrix

Front 68

ad-bc is...

Back 68

determinant

Front 69

test for invertibility

Back 69

ad-bc != 0

Front 70

If A is an invertible n*n matrix for each b in R^n

Back 70

the equation Ax=b has a unique solution

x= (A^-1)b
A((A^-1)b=b
b=b (true!)

Front 71

invertible matrix is row equivalent to an identity matrix how?

Back 71

by watching the row reduction of A to I

Front 72

a square matrix A is invertible iff A is row equivalent to an identity matrix or after you perform elementary row operations that reduce A to In

Back 72

memorize statement

Front 73

IMT
A is row

Back 73

equivalent to the n*n identity matrix

Front 74

IMT
A has

Back 74

n pivot positions

Front 75

IMT
The equation Ax=0

Back 75

has only the trivial solution

Front 76

IMT
The columns

Back 76

of A form a linearly independent set

Front 77

IMT
The linear transformation x |--> Ax

Back 77

is one to one

Front 78

IMT
The equation Ax=b

Back 78

has at least one solution for each b in R^n

Front 79

IMT
The columns

Back 79

of A span R^n

Front 80

IMT
The linear transformation x |--> Ax

Back 80

maps R^n onto R^n

Front 81

A^T is

Back 81

an invertible matrix

Front 82

There is an n*n matrix D such that

Back 82

AD=I