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.

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2
u \notequal \alpha{v}
u \notequal \alpha{v} + \beta{w}

This means u is linearly independent with v or w.

Definition: u, v, w are linearly independent.

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2
av + bw + cu = 0
u \notequal av + bw

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

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2
i = [1,0]
j = [0,1]

are basis vectors of the x-y coordinate system.


Reference
https://www.3blue1brown.com/topics/linear-algebra