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| geometrically represented by a directed lign segment with its initial point at the origin and its terminal point at (x1,x2) |
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| how we represent the vector x. Also x1 and x2 are called the components of the vector x. |
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| we say two vectors x=(x1,x2) and u=(u1,u2) are equal if and only if x1=u1 and x2=u2 |
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the sum of two vectors x=(x1,x2) and u=(u1,u2) is defined as the vector
x+u=(x1+u1, x2+u2) |
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| vector scalar multiplication |
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to multiply a vector x=(x1,x2) by a scalar c we multiply each component by c.
c1*x=(cx1,cx2) |
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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) |
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| u+v is a vector in the plane |
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| commutative property of addition |
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| associative property of addition |
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| additive inverse property |
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| cu is a vector in the plane |
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| closure under scalar multiplication |
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| associative property of multiplication |
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| multiplicative identity property |
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| vector operations extended to higher dimensions |
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| 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 |
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| if and only if their components are equal |
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| standard operations in Rn |
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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) |
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| the negative of vector u in Rn |
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| difference between two vectors u and v |
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| u-v=(u1-v1,u2-v2,...un-vn) |
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| commutative property of addition |
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| associative property of addition |
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| additive identity property |
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| additive inverse property |
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| closure under scalar multiplication |
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| associative property of multiplication |
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| multiplicative identity property |
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| properties of additive identity and additive inverse |
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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
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| vector x is called a linear combination of the vectors v1,v2,...vn if |
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