What is Rank of a family?

Example 19.16 (Computing a rank by elimination)

Rank of $\bigl((1,2,3), (2,3,4), (3,4,5), (1,1,1)\bigr)$ in $\R^3$. The span is unchanged when one subtracts from a vector a combination of the others (both families span the same combinations): replace $(2,3,4)$ by $(2,3,4) - (1,2,3) = (1,1,1)$ and $(3,4,5)$ by $(3,4,5) - (1,2,3) = (2,2,2)$. The span is now $\operatorname{Vect}\bigl((1,2,3), (1,1,1), (2,2,2), (1,1,1)\bigr) = \operatorname{Vect}\bigl((1,2,3), (1,1,1)\bigr)$, and these two vectors are not proportional: the rank is $2$. This subtract-and-discard procedure is systematized as Gaussian elimination in Chapter 22.

Example 19.17 (Summing by concatenation)

Take, in $\R^3$,

The sum $F + G$ is spanned by the concatenated family of all three generators, and

shows the third is redundant: $F + G = F$, of dimension $2$ — equivalently $G \subseteq F$, which the relation displays. Grassmann confirms: $\dim(F \cap G) = 2 + 1 - 2 = 1 = \dim G$. Sums are computed by concatenating generators and then reducing the pile by the rank algorithm; no new technique is ever needed.