Linear Algebra Test #1 - General Terms And Equations

Front 1

row reduction allows us to :

Back 1

1) Swap rows
2) Scale rows
3) Combine rows

Front 2

Echelon form

Back 2

1 pivot per row

Nothing to the left of a pivot in a row

Each pivot is further to the right than the pivot above

Front 3

A pivot on the right most column means

Back 3

There is no solution.

Front 4

reduced echelon form

Back 4

all pivots are '1'
only zeros in the same columns as pivots

Front 5

trivial solution is

Back 5

all zeros

Front 6

homogeneous system is

Back 6

set equal to zero

Front 7

All homogeneous solutions have

Back 7

-the origin as a solution
-can only have a non-trivial solution of you have a free variable
-the solution can be scaled and it is still a solution
-adding 2 solutions, also, yields a solution

Front 8

the general solution =

Back 8

homogenous solution + particular solution

Front 9

# of free variables =

Back 9

# of degrees of freedom & # of dimensions in answer

Front 10

length

Back 10

sqrt of all entries squared

Front 11

dot product

Back 11

the sum of all the products of corresponding vector entries

Front 12

expeted value

Back 12

probability vector (dot) outcome vector

Front 13

v (dot) w , is also = to

Back 13

[v][w]cos (theta)

Front 14

v (dot) v =

Back 14

[v]^2

Front 15

perpendicular=

Back 15

orthogonal= dot product is zero

Front 16

a convex combination

Back 16

xv+(1-x)w

Front 17

the 3 properties of a sub set that are not necessarily inherited by a vector space:

Back 17

1-includes the zero vector
2-closed under addition
3- closed under multiplication

Front 18

if your matrix has free variables, then you set is linearly :

Back 18

DEPENDENT

Front 19

of your matrix does not have free variables, then your set is linearlly:

Back 19

independent

Front 20

when testing to see if a set or sub set is a vector space , be sure to test using -

Back 20

scalar multiples of zero and negative 1

Front 21

Span: ?

Back 21

The set of all linear combinations of those vectors

Front 22

when writing the span

Back 22

include only the vectors of the free variables, surrounded by squiggle brackets

Front 23

the only time when span is not infinite-

Back 23

it is zero

Front 24

linear independence:

Back 24

a set of vectors is linearly independent if non of the vectors in the set is a linear combination of the other vectors. The opposite is linear DEPENDENCE.

Front 25

can you make a linearly dependent set into an indep set by adding a vector?

Back 25

NO

Front 26

can you make a lin dep set into an indep set by subtracting a vector

Back 26

YES

Front 27

if S1 is a subset of s

Back 27

1- if s1 is linearly indep so is the bigger set s
2- if the bigger set is linearly dep, then so is its small ser s1

Front 28

can you add a vector to a lin indep set and make it dep?

Back 28

YES

(so long as the vector you add in wasnt in the span of the origoinal vecotr)

Front 29

Basis

Back 29

A set of vectors that are:

1) Linearly independent

2) Span all of V

Front 30

How to test for a basis?

Back 30

1) pivots in all columns - (no free variables) = independent

2) pivots in all rows = spans whole set

** a little bonus to remember : given a vector space, there is exactly 1 way to express a vector (in the set of V) as a linear combination of the vectors from the basis.

Front 31

angle between 2 vectors w and v:

Back 31

cos(theta)= (w dot v)/ ([w][v])

Front 32

What is a pivot element in the context of linear algebra?

Back 32

The pivot element is an element on the left hand side of a matrix that you want elements above and below to be the value zero.

Front 33

Are the vectors:

v1 = (2,2,0)

v2 = (1,-1,1)

v3 = (4,2,-2)

linearly independent?

Back 33