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Definition
geometrically represented by a directed lign segment with its initial point at the origin and its terminal point at (x_{1},x_{2}) 


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ordered pair (x_{1,}x_{2}) 

Definition
how we represent the vector x. Also x_{1 }and x_{2 }are called the components of the vector x. 


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Definition
we say two vectors x=(x_{1},x_{2}) and u=(u_{1},u_{2}) are equal if and only if x_{1}=u_{1} and x_{2}=u_{2} 


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Definition
the sum of two vectors x=(x_{1},x_{2}) and u=(u_{1},u_{2}) is defined as the vector
x+u=(x_{1}+u_{1}, x_{2+}u_{2}) 


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vector scalar multiplication 

Definition
to multiply a vector x=(x_{1},x_{2}) by a scalar c we multiply each component by c.
c_{1}*x=(cx_{1},cx_{2}) 


Term

Definition
as a consequence of vector addition and scalar multiplication, we can now define the subtraction of two vectors x=(x_{1},x_{2}) and u=(u_{1},u_{2}) as
x+(u)=xu=(x_{1}u_{1},_{ }x_{2}u_{2}) 


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Definition


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u+v is a vector in the plane 

Definition


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Definition
commutative property of addition 


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Definition
associative property of addition 


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Definition
additive inverse property 


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cu is a vector in the plane 

Definition
closure under scalar multiplication 


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Definition


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Definition


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Definition
associative property of multiplication 


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Definition
multiplicative identity property 


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Definition
vector operations extended to higher dimensions 


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Definition
represents a vector in nspace and has the form (x_{1},x_{2,}x_{3},...x_{n}). We can either view these ntuples as points in R^{n}with the x_{i}'s as its coordinates or as a vector x=(x_{1},x_{2,...}x_{n}) with x_{i}'s as its components 


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Definition
if and only if their components are equal 


Term
standard operations in R^{n} 

Definition
let u=(x_{1},x_{2}...,u_{n}) and v=(v_{1},v_{2},...v_{n}) be vectors in R^{n} and c be a real number. Then the sum of u and v is defined as the vector
u+v=(u_{1}+v_{1,}u_{2}+v_{2},...u_{n}+v_{n})
and the scalar multiple of u by c is defined as the vector
cu=(cu_{1},cu_{2},...cu_{n}) 


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the negative of vector u in R^{n} 

Definition
u=(u_{1},u_{2},...u_{n}) 


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difference between two vectors u and v 

Definition
uv=(u_{1}v_{1},u_{2}v_{2},...u_{n}v_{n}) 


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Definition


Term

Definition
commutative property of addition 


Term

Definition
associative property of addition 


Term

Definition
additive identity property 


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Definition
additive inverse property 


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Definition
closure under scalar multiplication 


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Definition


Term

Definition


Term

Definition
associative property of multiplication 


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Definition
multiplicative identity property 


Term
properties of additive identity and additive inverse 

Definition
let v be a vector in R^{n}, 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 v_{1},v_{2},...v_{n }if 

Definition
x=c_{1}v_{1}+c_{2}v_{2}+...c _{n}v_{n} 

