7. Basis and Dimension of Subspace

1. Span

Verb : A Set of vectors is said to span a space if the set of all their linear combinations is the space

 

Noun : $span(v_{1}, v_{2}) = \{ c_{1} v_{1} + c_{2} v_{2}, \forall c_{1}, c_{2} \in \mathbb{R} \}$

 

Note : vectors that span a space are not necessarily independent

 

Fact :  columns of $A$ span $C(A)$, special solutions of $A$ span $N(A)$

 

 

 

2. Basis

A Basis of a space is a set of linearly independent vectors that span the space (무수히 많은 basis가 존재할 수 있음)

 

Note : For a given basis, there is only one linear combination of the basis vectors that can represent a vector in its span

 

How to find a basis of $C(A)$?

1. Identify pivot columns (via elimination)
2. The columns of $A$ (not the matrix obtained after elimination. So, original $A$) having pivots are a basis of $C(A)$

 

Note : The columns of any invertible $(n \times n)$ matrix give a basis for $\mathbb{R}^{n}$ (full column rank 이므로)

 

 

 

3. Dimension

The dimension of a space is the number of vectors in every basis

 

For $(m \times n)$ matrix $A$,

 

$dim(C(A)) = r(A), dim(N(A)) = n - r(A)$