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Linear Algebra test 2
material from 4.1
34
Mathematics
Undergraduate 2
10/24/2016

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Term
vector in a plane
Definition
geometrically represented by a directed lign segment with its initial point at the origin and its terminal point at (x1,x2)
Term
ordered pair (x1,x2)
Definition
how we represent the vector x. Also xand xare called the components of the vector x.
Term
equal vector
Definition
we say two vectors x=(x1,x2) and u=(u1,u2) are equal if and only if x1=u1 and x2=u2
Term
vector addition
Definition

the sum of two vectors x=(x1,x2) and u=(u1,u2) is defined as the vector

x+u=(x1+u1, x2+u2)

Term
vector scalar multiplication
Definition

to multiply a vector x=(x1,x2) by a scalar c we multiply each component by c.

c1*x=(cx1,cx2)

Term
vector subtraction
Definition

as a consequence of vector addition and scalar multiplication, we can now define the subtraction of two vectors x=(x1,x2) and u=(u1,u2) as

x+(-u)=x-u=(x1-u1, x2-u2)

Term
negative of vector x
Definition
(-1)x=-x
Term
u+v is a vector in the plane
Definition
closure under additon
Term
u+v=v+u
Definition
commutative property of addition
Term
(u+v)+w=u+(v+w)
Definition
associative property of addition
Term
u+0=0
Definition
additive inverse property
Term
cu is a vector in the plane
Definition
closure under scalar multiplication
Term
c(u+v)=cu+vu
Definition
distributive property
Term
(c+d)u=cu+du
Definition
distributive property
Term
c(du)=(cd)u
Definition
associative property of multiplication
Term
1(u)=u
Definition
multiplicative identity property
Term
Rn
Definition
vector operations extended to higher dimensions
Term
ordered n-tuple
Definition
represents a vector in n-space and has the form (x1,x2,x3,...xn). We can either view these n-tuples as points in Rnwith the xi's as its coordinates or as a vector x=(x1,x2,...xn) with xi's as its components
Term
two vectors are equal...
Definition
if and only if their components are equal
Term
standard operations in Rn
Definition

let u=(x1,x2...,un) and v=(v1,v2,...vn) be vectors in Rn and c be a real number. Then the sum of u and v is defined as the vector

u+v=(u1+v1,u2+v2,...un+vn)

and the scalar multiple of u by c is defined as the vector

cu=(cu1,cu2,...cun)

Term
the negative of vector u in Rn
Definition
-u=(-u1,-u2,...-un)
Term
difference between two vectors u and v
Definition
u-v=(u1-v1,u2-v2,...un-vn)
Term
u+v is a vector in Rn
Definition
closure under addition
Term
u+v=v+u
Definition
commutative property of addition
Term
(u+v)+w=u+(v+w)
Definition
associative property of addition
Term
u+0=u
Definition
additive identity property
Term
u+(-u)=0
Definition
additive inverse property
Term
cu is a vectory in R
Definition
closure under scalar multiplication
Term
c(u+v)=cu+vu
Definition
distributive property
Term
(c+d)u=cu+cd
Definition
distributive property
Term
c(du)+(cd)u
Definition
associative property of multiplication
Term
1(u)=u
Definition
multiplicative identity property
Term
properties of additive identity and additive inverse
Definition

let v be a vector in Rn, and let c be a scalar. Then the following properties are true:

1. the additive property is unique, if v+u=v, u=0

2. the additive inverse of v is unique, if v+u=0, u=-v

3. 0*v=0

4. c*0=0

5. if cu=0, them c=0 or u=0

6. -(-v)=v

 

Term
vector x is called a linear combination of the vectors v1,v2,...vif
Definition
x=c1v1+c2v2+...c nvn
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