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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
✞
Lecture
✝
1
2
3
☎
01 – Vectors ✆
4
Vectors
Vector addition
Scalar multiplication
5
6
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
1/7
MATH1151 (Algebra)
Vector addition
L01 – Vectors
Session 1, 2014
2/7
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)
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
✞
Lecture
✝
1
2
3
☎
01 – Vectors ✆
4
Vectors
Vector addition
Scalar multiplication
5
6
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
1/7
MATH1151 (Algebra)
Vector addition
L01 – Vectors
Session 1, 2014
2/7
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)