Vector span
It’s extending the unit vector
idea.
Refer to famous visualisation of 3Blue1Brown’s video: Linear combinations, span, and basis vectors
R²
and R³
R²
means a Real numbers 2D plane
.
Usually the X/Y Axes plane
is this one.
R³
means Real numbers 3D plane
.
Usually the X/Y/Z Axes plane
.
Linear combinations (Vector Addition)
DEFINITION: The sum of
cv
anddw
is a linear combination ofv
andw
.
Linear combinations
means to add vectors together: v₁ + v₂ + v₃.....
to get a new vector. Simple like that.
Span of vectors
It’s the Set of all the
linear combinations
of a number vectors.
# v, w are vectors
span(v, w) = R²span(0) = 0
One vector
with a scalar
, no matter how much it stretches or shrinks, it ALWAYS on the same line, because the direction or slope is not changing. So ONE VECTOR'S SPAN IS A LINE.
Two vector
with scalars
, we then COULD change the slope! So that we could get to any position that we want in the 2D plane, i.e., R².
Exceptions:
span(0) = 0
, it only stay at origin.v = w
, if two vectors are the same, orcollinear
, then it's still ONE vector.
Linearly dependent & independent
Linear dependence
: two vectors areCOLLINEAR
, means on the same line.Linear independence
: two vectors areNOT COLLINEAR
, means they're not on the same line.
Vectors (2, 3)
and (4, 6)
are the SAME VECTOR!
Because (4,6) = 2*(2,3)
, so it's just a scaled version of the first vector.
So we say the vectors (2, 3)
and (4, 6)
are DEPENDENT
, because they're COLLINEAR
.
Other than that, any two vectors are INDEPENDENT
, if they're not NOT COLLINEAR.
List of some linear combinations
Let’s list some vector combinations
:
- Zero Vector:
span(0) = 0
. - One vector:
span(v) = a line
. - Two vector:
span(v₁, v₂) = R²
, if they're not collinear. - Three vector or more:
span(v₁, v₂, v₃...) = R²
. Other than two vectors, are allREDUNDANT
.
In another word:
IF ANY TWO VECTORS ARE INDEPENDENT, THEN OTHERS ARE ALL DEPENDENT.
How to calculate a linear combination's independency
Refer to Adam Panagos: Linear Algebra Example Problems — Linearly Independent Vectors #1
[Refer to TheTrevTutor: [Linear Algebra] Linear Independence and Bases](https://www.youtube.com/watch?v=OLqc_rt7abI)
Refer to Khan lecture: Span and linear independence example
It’s important for knowing if a
linear combination
can fill out a plane or space.
For example, if two vectors aren't independent, then it's just one vector, and can only draw a line. If three vectors aren't independent, then they're just two vectors, one is redundant, so they can only fill out a 2D plane instead of a 3D space.
A linear combination
is independent, iff it could satisfy this equation:
c₁v₁ + c₂v₂ + c₃v₃ .... = 0
c..
means the scalar for each vector, and you could change the scalar to any number, positive or negative.
Note that: c ≠ 0
, and vectors are not all zeros.
Assume that there’s a linear combination of two vectors v₁ + v₂ + v₃
,
with scalars it could be c₁v₁ + c₂v₂ + c₃v₃
.
To verify whether it's dependent or independent,
we assume c₁v₁ + c₂v₂ + c₃v₃= (0,0,0)
and solve for c₁, c₂, c₃
:
- it’s independent <=> if
c₁ = c₂ = c₃ = 0
all are zeros - it’s dependent <=> If
c₁, c₂, c₃
at least one is NON-ZERO number
Independent Example:
Dependent Example:
-