finals: sections 1.1 - graph and adjacency matrices

Question 1

what is a linear equation?

Answer 1

a1x1 + a2x2 + … + anxn = b
where
a1, a2, …, an are real or complex numbers
b is a real or complex number
x1, x2, …, xn are variables

Question 2

when are two linear systems equivalent?

Answer 2

if they have the same solution set

Question 3

what is a solution set of a linear system?

Answer 3

the set of all solutions that makes all of the linear equations in the system true

Question 4

a system of linear equations has : (solutions)

Answer 4

• exactly one solution
• infinitely many solutions
• no solution

Question 5

when is a system of linear equations consistent?

Answer 5

when the system has at least one solution

Question 6

when is a system of linear equations inconsistent?

Answer 6

when the system has no solution

Question 7

what is a coefficient matrix?

Answer 7

the coefficients attached to the variables are put into a matrix

Question 8

what is an augmented matrix?

Answer 8

a coefficient matrix with an added column of constants (that are the entries in the b vector)

Question 9

what is the size of a matrix?

Answer 9

of rows x # of columns (rows x cols)

Question 10

what are the three elementary row operations?

Answer 10

1. Replacement: replace one row by the sum of itself and a multiple of another row
2. Interchange: interchange two rows
3. Scaling: multiply all entries in a row by a nonzero constant

Question 11

when are two matrices row equivalent?

Answer 11

two matrices are row equivalent if there a exists a sequence of elementary row operations that transforms one matrix into another

Question 12

if the augmented matrices of two linear systems are row equivalent __________________

Answer 12

then the two systems have the same solution set

Question 13

what are the two fundamental questions about a linear system?

Answer 13

1) is the system consistent? that is, does it have at least one solution?
2) if a solution exists, is it the only one?

Question 14

what is the leading entry of a row mean?

Answer 14

the leftmost nonzero entry in a nonzero row

Question 15

what are the conditions for a matrix to be in echelon form?

Answer 15

1) all nonzero rows are above any rows of zeros
2) each leading entry of a row is a column to the right of the leading entry of the row above it (lower triangle of zeros)
3) all entries in the column below a leading entry are zeros (lower triangle of zeros)

Question 16

what are the conditions for a matrix to be in reduced row echelon form?

Answer 16

it must satisfy the conditions to be in echelon form and these two conditions:
1) the leading entry in each nonzero row is 1
2) each leading 1 is the only nonzero entry in its column

Question 17

is the reduced row echelon form (RREF) of a matrix unique?

Answer 17

yes

Question 18

is the echelon form of a matrix unique?

Answer 18

no

Question 19

what is a pivot position in a matrix?

Answer 19

it is a location in a matrix A that corresponds to a leading 1 in the RREF of A.

Question 20

what is a pivot column?

Answer 20

a column of A that contains a pivot position

Question 21

what is span? (use span{v1, …, vp})

Answer 21

span{v1, …, vp} is the set of all the linear combinations of v1, …, vp

Question 22

what is a vector equation?

Answer 22

x1[vector] + x2[vector] + … + xp[vector] = [b vector]

Question 23

asking if a vector b is in span{v1, …, vp} is the same as asking _______________

Answer 23

does the augmented matrix [v1 … vp b] have a solution?

Question 24

the zero vector is ALWAYS in ________

Answer 24

span

Question 25

the equation Ax = b has a solution if and only if __________

Answer 25

b is a linear combination of the columns of A

Question 26

what is the definition of a graph G? (graphs and adjacency matrices)

Answer 26

a mathematical structure used to model pairwise relations between objects. a graph is defined as a pair G = (V, E)

Question 27

in a graph G = (V, E), what is V?

Answer 27

a set of vertices (or nodes)

Question 28

in a graph G = (V, E), what is E?

Answer 28

a set of edges, or links, which are two element subsets of fV

Question 29

what is an undirected graph?

Answer 29

edges have no direction

Question 30

what is a directed graph?

Answer 30

edges have a direction

Question 31

what is a weighted graph?

Answer 31

each edge has an associated weight

Question 32

what is an unweighted graph?

Answer 32

edges do not have weights

Question 33

what is the definition of an adjacency matrix?

Answer 33

it is a square matrix used to represent a graph G. the elements of A are used to indicate whether pairs of vertices are adjacent (connected to each other) or not in the graph

Question 34

what does a 1 mean in an adjacency matrix?

Answer 34

the two nodes are connected to each other

Question 35

what does a 0 mean in an adjacency matrix?

Answer 35

the two nodes are NOT connected to each other