# Linear Algebra - Rank

The rank of a set S of vectors is the dimension of Span S written:

• rank S

rank = dim Span

Any set of D-vectors has rank at most |D|.

If rank(S) = len(S) then the vectors are linearly dependent (otherwise you will get len(S) > rank (S)).

## 3 - Matrix

For a linear function Matrix f(x) = $$\text{dim lm f = dim Col A = rank A}$$ where:

## 4 - Example

### 4.1 - No empty set of vectors

The vectors [1, 0, 0], [0, 2, 0], [2, 4, 0] are linearly dependent. Therefore their rank is less than three. First two of these vectors form a basis for the span of all three, so the rank is two.

### 4.2 - empty set of vectors

The vector space Span {[0, 0, 0]} is spanned by an empty set of vectors. Therefore the rank of {[0, 0, 0]} is zero