Lecture Chapter 1

Question 1

What is a solution set?

Answer 1

The set of all possible solutions of a linear system

Question 2

What are equivalent systems?

Answer 2

They have the same solution set

Question 3

What are the possibilities of solutions of a system

Answer 3

either 1, zero or infinitely many solutions

Question 4

What does consistent mean? inconsistent?

Answer 4

consistent : one or infinitely many solutions

inconsistent : no solution

Question 5

How many rows and columns has a mxn matrix?

Answer 5

m rows, n columns

Question 6

three characteristics of echelon form

Answer 6

• all nonzero rows are above any rows of all zeros
• each leading entry is on the right of the one above
• all entries below a leading entry are zeros

Question 7

three characteristics of reduced echelon form

Answer 7

• echelon form conditions
• the leading entry in each nonzero row is 1
• each leading 1 is the only nonzero in its column

Question 8

What is a pivot position

Answer 8

leading 1 in the reduced echelon form

Question 9

What are row equivalent matrices

Answer 9

if row operations can transform one into the other

the two systems are equivalent (same solution set)

Question 10

Can a matrix be row equivalent to multiple reduced echelon matrices

Answer 10

no, only one

Question 11

what characterizes a consistent linear system

Answer 11

no [0 0 0 … 0 / b] in the echelon form of the augmented matrix

Question 12

What is a linear combination

Answer 12

vector y = c1v1 + c2v2 + … + cpvp

linear combination of v1,v2,…,vp with weights c1,c2,…,cp

Question 13

How do you find the solution of a vector equation x1a1 + x2a2 + … + xnan = b

Answer 13

the equation has the same solution set as the linear system with augmented matrix [a1 a2 … an / b]

Question 14

What is Span {v1 v2 … vp} ?

Answer 14

The subset of Rn spanned by v1, v2, … , vp

The set of all linear combinations of v1, v2, … , vp (collection of vectors)

Question 15

What is Ax

Answer 15

Linear combination of the columns of A using the corresponding entries in x as weights

Question 16

How do you find the solution of Ax=b?

Answer 16

same solution set as
x1a1 + x2a2 + … + xnan = b
[a1 a2 … an b]

Question 17

What statements are equivalent to ‘for each b in Rm, Ax=b has a solution’

Answer 17

• each b in Rm is a linear combination of the columns of A
• the columns of A span Rm
• A has a pivot position in every row

Question 18

A(u+v) = Au + Av       
A(cu) = c(Au)

Question 19

What is a homogeneous linear system?

Answer 19

Ax=0

Question 20

Two properties of Ax=0 (homogeneous linear system)

Answer 20

• always has at least one solution : trivial solution : zero vector 0
• if Ax=0 has a nontrivial solution = equation has at least one free variable

Question 21

parametric vector equation of a line through the origin

Answer 21

x = su

Question 22

parametric vector equation of a plane through the origin

Answer 22

x = su + tv

Question 23

parametric vector equation of a line not through the origin

Answer 23

x = su + p

Question 24

parametric vector equation of a plane not through the origin

Answer 24

x = su + tv + p

Question 25

When is a set of vectors linearly independent?

Answer 25

When the vector equation x1v1 + x2v2 + ... + xnvn = 0 | has only the trivial solution

Question 26

When is a set of vectors linearly dependent? What is the linear dependence relation?

Answer 26

When there exists weights not all zero such that c1v1 + c2v2 + ... + cpvp = 0 (linear dependence relation)

Question 27

When are the columns of A linearly independent?

Answer 27

If Ax=0 has only the trivial solution

Question 28

What can you say about a set containing one vector linearly dependent?

Answer 28

It's the zero vector

Question 29

What can you say about a set containing two vectors linearly dependent?

Answer 29

The vectors are a multiple of one another

Question 30

What can you say about a set containing two or more vectors linearly dependent?

Answer 30

At least one is a linear combination of the others

Question 31

What can you say about a set containing more vectors than there are entries in each vector

Answer 31

The set is linearly dependent

Question 32

What can you say about a set containing the zero vector

Answer 32

The set is linearly dependent

Question 33

What can you say about a transformation T from Rn to Rm

Answer 33

- assigns to each x in Rn a T(x) in Rm - Rn domain - Rm codomain - T(x) in Rm is the image of x in Rn - The set of all images is the range of T

Question 34

What is T(x)=Ax

Answer 34

matrix transformation

Question 35

Three properties to show a transformation T is linear

Answer 35

- T(u+v) = Tu + Tv - T(cu) = cT(u) - T(0) = 0

Question 36

What is an identity matrix

Answer 36

1 in the main diagonal, zeros elsewhere

Question 37

When is T : Rn-->Rm onto Rm?

Answer 37

if each b in Rm is the image of at least one x in Rn

Question 38

When is T : Rn-->Rm one-to-one?

Answer 38

if each b in Rm is the image of at most one x in Rn - -> T(x)=0 only has the trivial solution - -> the columns of A are linearly independent

Question 39

When does T maps Rn onto Rm?

Answer 39

When the columns of A span Rm