Final Exam

Question 1

1.1 A system of linear equations has how many soluntions

Answer 1

no solution, exactly one solution, or infinite solutions

Question 2

1.1 consistent

Answer 2

system has a solution

Question 3

1.1 inconsistent

Answer 3

system has no solution

Question 4

1.2 Free variable

Answer 4

no pivot in this row x3 = x3

Question 5

1.2 Existence and Uniqueness Theorem: A system is consistent only if

Answer 5

the augmented column is not a pivot column

Question 6

1.2 If a linear system is consistent then

Answer 6

it has a unique solution when there’s no free variables, it has infinite solutions when there is at least 1 free variable

Question 7

1.3 Parallelogram Rule

Answer 7

0, u, and v are points on a plane. Then the 4th point is u+v

Question 8

1.3Linear combination

Answer 8

y = c1v1 + … + cpvp

Question 9

1.3 Span

Answer 9

set of all linear combinations of v1….vp

Question 10

1.4 Theorem 3: The equation Ax=b has the same solution set as ____

Answer 10

x1a1 + ……xpap

Question 11

Ax = b has a solution( is consistent) only if ____

Answer 11

b is a linear combination of the columns of A.

Question 12

1.5 Homogeneous

Answer 12

system can be written in the form Ax =0

Question 13

1.5 Trivial Solution

Answer 13

solution where x =0

Question 14

1.5 Nontrivial solution

Answer 14

a nonzero vector x that makes Ax =0

Question 15

1.6 Linearly Independent

Answer 15

a set of vectors is linearly independent if x1v1+…xpvp =0

Question 16

1.7 Linearly Dependent

Answer 16

A set of vectors is linearly dependent if there are nonzero constants that make c1vi+…cpvp=0

Question 17

1.7 Theorem: The columns of A are linearly independent if what

Answer 17

The equation Ax=0 has only the trivial solution (where x=0)

Question 18

1.7 A set of 2 vectors is linearly dependent if what

Answer 18

if at least one of the vectors is a multiple of the other.

Question 19

1.7 A set is linearly independent if what?

Answer 19

if the vectors are not multiples of one another

Question 20

1.7 Characterization of Linearly Dependent sets: An set of 2+ vectors is linearly dependent if

Answer 20

if one of the vectors is a linear combination(y=c1v1) of the others or if v1=0.

Question 21

1.7 If a set contains more vectors than there are entries in each vector, then…

Answer 21

the set is linearly independent

Question 22

1.8 domain

Answer 22

The set R^n

Question 23

1.8 codomain

Answer 23

the set R^m

Question 24

1.8 image

Answer 24

the vector T(x) in R^m

Question 25

1.8 range

Answer 25

the set of all images T(x)

Question 26

1.8 if T is a linear combination then T(cu+dv)=?

Answer 26

cT(u)+dT(v)

Question 27

1.8 A transformation T is linear if: (hint 2)

Answer 27

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

Question 28

1.9 Let T: R^n-> R^m be a linear transformation. Then there exists a unique matrix A such that

Answer 28

T(x) = Ax

Question 29

1.9 Onto

Answer 29

pivot in every row

Question 30

1.9 one to one

Answer 30

pivot in every column

Question 31

1.9 A linear transformation is one to one only if

Answer 31

T(x) = 0 has the trivial solution (x=0)

Question 32

1.9 T maps R^n onto R^m only if

Answer 32

the columns of A span R^m

Question 33

1.9 T is one to one only if

Answer 33

the columns of A are linearly independent

Question 34

2.1 (matrix multiplication) If A is an mxn matrix and B is an nxp matrix, the the product of AP is what

Answer 34

the mxp matrix whose columns are [Ab1.....Abp]

Question 35

2.1 Each column of AB is what

Answer 35

a linear combination of the columns of A using the constants from the corresponding column of B

Question 36

2.1 (A^T)^T =?

Answer 36

A

Question 37

2.1 (A+B)^T = ?

Answer 37

A^T + B^T