Linear combinations, span, and basis vectors
Combinations
1 | u = \alpha{v} + \beta{w} |
u
is the linear combination of v
and w
, \alpha
and \beta
are scaler.
This is also means u is linearly dependent with v
or w
.
1 | u \notequal \alpha{v} |
This means u is linearly independent with v
or w
.
Definition: u, v, w are linearly independent.
1 | av + bw + cu = 0 |
Span
The span of v
and w
is the set of all their linear combinations.
For 3-Dimension
The basis of a vector space is a set of linearly independent vectors that span the full space.
Basis vectors
1 | i = [1,0] |
are basis vectors of the x-y coordinate system.
Reference
https://www.3blue1brown.com/topics/linear-algebra