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Vectors

Topic and contents

Vectors
Definition (Vectors in Rn )
For any positive integer n, a vector a ∈ Rn is an n-tuple of real numbers, that is an ordered list of n real numbers

School of Mathematics and Statistics
MATH1151 – Algebra

(a1 , a2 , a3 , . . . , an−1 , an )
Notation: a ∈ Rn , vector a, by hand a , ∈ is an element of
˜
n)
Example (Vectors in R
√
u = (1, 2), v = (2, 1) ∈ R2 z = (0, π, 3.2, e, 2, 4) ∈ R6

A/Prof Rob Womersley
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Lecture
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01 – Vectors ✆
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Vectors
Vector addition
Scalar multiplication

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w = (1, 0, −1) ∈ R3

ASX 200 share prices, x ∈ R200

x = (−1, 0, 1, 2, 3) ∈ R5

Distance between vectors
Equality of vectors
Logic

0 = (0, 0, 0, 0, 0) ∈ R5

y = (−1, 0, 1, 2) ∈ R4

0 = (0, 0, 0, 0) ∈ R4

Notes
Order matters http://www.asx200.com/ MATH1151 (Algebra)

L01 – Vectors

Session 1, 2014

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MATH1151 (Algebra)

Vector addition

L01 – Vectors

Session 1, 2014

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Scalar multiplication

Vector addition

Scalar multiplication
Definition (Scalar multiplication)

Definition (Vector addition)

Scalar multiplication (multiplication of a vector by a real number) λ ∈ R, a ∈ Rn

Addition of two vectors a, b ∈ Rn : a + b = (a1 , a2 , . . . , an ) + (b1 , b2 , . . . , bn )

λa = λ(a1 , a2 , . . . , an ) = (λa1 , λa2 , . . . , λan ) ∈ Rn

= (a1 + b1 , a2 + b2 , . . . , an + bn ) ∈ Rn

Notes
The result λa has the same number of elements as the vector a
Every component of the vector a is multiplied by the scalar λ

Notes
The vectors a and b and the result a + b all have the same number of elements. Example (Scalar multiplication)

Example (Vector addition) x = (−1, 0, 1, 2, 3),

y = (−1, 0, 1, 2),

x = (−1, 0, 1, 2, −3)
−3x = ?

x+y= ?

Not defined, as vectors do not have the same number of elements

x = (−1, 0, 1, 2, 3),

z = (−1, 2, −1, 3, −4),

(3, 0, −3, −6, 9) ∈ R5

x+z = ?

0x = ?

x + z = (−2, 2, 0, 5, −1)