Linear Algebra 1

Front 1

if A and B are any square n x n matrices, AB=BA

Back 1

false

Front 2

an augmented matrix whose last column contains only 0's is representative of a consistent linear system

Back 2

true

Front 3

if the vector 0 is contained in the set of vectors {v1,v2,…vn} then the collection of vectors must be linearly dependent

Back 3

true

Front 4

T(x) = x^3 is a linear transformation

Back 4

false

(u+v)^3 does not equal u^3 + v^3

Front 5

it is possible for a set of n vectors in R n+1 to be linearly dependent

Back 5

true

Front 6

span {v1,v2,…vn} is the set of all vectors which can be written as linear combinations of the vectors v1,v2,…vn

Back 6

true

Front 7

more columns than row entries

Back 7

means linearly dependent

Front 8

A linear combination of {v1,v2,…vn} is

Back 8

any expression of the form c1v1 +…cnvn where c1,…cn are scalars

Front 9

the set of vectors {v1,v2,…vn} is linearly INDEPENDENT

Back 9

the augmented matrix describes a system that has only the trivial solution

Front 10

A transformation T: Rn --> Rm is linear if

Back 10

for all u, v in Rn and all scalars C
T(u+v) = T(u) + T(v) and
T(cu) = cT(u)

Front 11

the span of a set of vectors {v1,v2,…vn} is

Back 11

the set of all possible linear combos of those vectors

Front 12

ax=0 is always consistent

Back 12

true

Front 13

a collection of n+1 vectors in Rn may be linearly independent

Back 13

false

Front 14

a 3x4 augmented matrix with 3 pivot points is representative of a linear system with a unique solution

Back 14

false

Front 15

{u, v, u-v} is a linearly dependent collection of vectors

Back 15

true

Front 16

Thm: following statements equivalent for A an m x n matrix

Back 16

1. for all b in Rm, ax=b has a solution
2. all b in Rm are lin combo of column vectors A
3. column vectors of A span Rm
4. A has a pivot position in each row

Front 17

linear system

Back 17

collection of one or more linear equations involving same variables

Front 18

consistent

Back 18

one or infinitely many solutions

Front 19

if augmented matrix of 2 lin. system are row equiv

Back 19

the 2 systems have same solution set

Front 20

pivot position

Back 20

location in A that correlates to a leading 1 in reduced echelon form

Front 21

subset of Rn spanned by v1…vp

Back 21

if v1, …vp are in Rn then the set of all lin combo of v1, ..vp is denoted by span {v1,…vp}

Front 22

if A is mxn matrix and x is in Rn then product of A and x

Back 22

is lin combo of columns of A using correlated entries in x as weights

Front 23

trivial solution

Back 23

zero solution

Front 24

nontrivial solution

Back 24

nonzero solution

Front 25

homogenous

Back 25

ax=0

Front 26

linear independent

Back 26

if vector eqn x1v1 + x2v2 = 0 has only the trivial solution

Front 27

linear dependent

Back 27

if there are weights c1…cp not all zero such that c1v1 +c2v2 = 0

Front 28

transformation linear iff

Back 28

T(u+v) = T(u) + T(v) and
T(cu) = cT(u)

Front 29

dependent if

Back 29

1. one vector is multiple of the other

2. more columns than rows

3. set has the zero vector

Front 30

homogenous eqn Ax=0 has nontrivial soln iff

Back 30

eqn has at least 1 free variable