Vectors And Linear Systems Of Equations

Question 1

Vector

Answer 1

a directed line segment corresponding to a displacement from one point A to another point B

Question 2

Vector addition

Answer 2

In general if u = [u1, u2] and v = [v1, v2] then we define
u + v = [u1 + v1, u2 + v2].

Question 3

Scalar multiplication

Answer 3

Given a vector v = [v1, v2] and c ∈ R, the scalar multiple cv is
cv = [cv1, cv2].

Question 4

R3

Answer 4

R3 is the set of all ordered triples of real numbers R3 = {(x,y,z) : x,y,z ∈ R}.

Question 5

Linear combination of vectors

Answer 5

A vector v is a linear combination of vectors v1 , · · · , vk
if and only if there are scalars c1, c2, · · · , ck such that v=c1v1 +c2v2 +···+ckvk

Question 6

A linear equation in the n variables x1, x2, · · · , xn

Answer 6

an equation that can be written in the form
a1x1 +a2x2 +···anxn = b.

Question 7

System of linear equations

Answer 7

a finite set of linear equations

Question 8

A solution to a system of linear equations

Answer 8

a vector that is simultaneously a solution to all equations in the system.

Question 9

Consistent

Answer 9

A system of equations is consistent if it has at least one solution

Question 10

Row echelon form

Answer 10

A matrix is in row echelon form if and only if any all-zero rows are at the bottom and in each non-zero row, the first non-zero entry (the leading entry) is to the left of any leading entries below it

Question 11

Row reduction

Answer 11

the process for reducing any matrix to row- echelon form, using elementary row operations

Question 12

reduced row echelon form

Answer 12

A matrix is in reduced row echelon form if and only if it is in row echelon form, the leading entry in each nonzero row is 1 and each column containing a leading 1 has 0s everywhere else

Question 13

Homogeneous

Answer 13

A system of linear equations is homogeneous if and only if the constant term in each equation is zero.

Question 14

Spanning set

Answer 14

Let S = {v1, v2,··· ,vk} be a set of vectors in Rn. S is a spanning set for Rn if and only if span(S) = Rn.

Question 15

f:D→Y

Answer 15

The set D is the domain, the set Y is the codomain.

Question 16

Domain

Answer 16

The domain is the set of input arguments for which the function can be evaluated.

Question 17

Argument

Answer 17

Each input is called an argument to the function.

Question 18

Range

Answer 18

The set of possible outputs is the range of the function and is a subset of the codomain.

Question 19

Maximal domain

Answer 19

the subset of R for which the formula makes sense and gives a result in the codomain of the function

Question 20

Graph of a function f : D → R, with D ⊂ R

Answer 20

The graph of a function f : D → R, with D ⊂ R, is
Graph(f) = {(x,y) ∈ R2|x ∈ D, y = f(x)}. This graph is a subset of R2.

Question 21

Graph of a function f : D → R, with D ⊂ R2

Answer 21

The graph of afunctionf:D→R,withD⊂R2 is
Graph(f) = {(x,y,z) ∈ R3|(x,y) ∈ D, z = f(x,y)}
This graph is a subset of R3